Class preserving automorphisms of finite p-groups

CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II arXiv:1406.7365v1 [math.GR] 28 Jun 2014 MANOJ K. YADAV Abstract. Let G be a finite group minimally generated by d(G) elements and Autc (G) denote the group of all (conjugacy) class-preserving automorphisms of G. Continuing our work [Class preserving automorphisms of finite p-groups, J. London Math. Soc. 75(3) (2007), 755-772], we study finite p-groups G such that | Autc (G)| = |γ2 (G)|d(G) , where γ2 (G) denotes the commutator subgroup of G. If G is such a p-group of class 2, then we show that d(G) is even, 2d(γ2 (G)) ≤ d(G) and G/ Z(G) is homocyclic. When the nilpotency class of G is larger than 2, we obtain the following (surprising) results: (i) d(G) = 2. (ii) If |γ2 (G)/γ3 (G)| > 2, then | Autc (G)| = |γ2 (G)|d(G) if and only if G is a 2-generator group with cyclic commutator subgroup, where γ3 (G) denotes the third term in the lower central series of G. (iii) If |γ2 (G)/γ3 (G)| = 2, then | Autc (G)| = |γ2 (G)|d(G) if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian commutator subgroup of order at most 8. As an application, we classify finite nilpotent groups G such that the central quotient G/ Z(G) of G by it’s center Z(G) is of the largest possible order. For proving these results, we introduce a generalization of Camina groups and obtain some interesting results. We use Lie theoretic techniques and computer algebra system ‘Magma’ as tools. 1. Introduction This paper is devoted to the study of finite p-groups admitting maximum number of classpreserving automorphisms. An automorphism α of a group G is called class-preserving if α(x) ∈ xG for all x ∈ G, where xG denotes the conjugacy class of x in G. The set of all classpreserving automorphisms of G, denoted by Autc (G), forms a normal subgroup of the group of all automorphisms of G, and contains Inn(G), the group of all inner automorphisms of G. For a finite group G minimally generated by d elements x1 , x2 , . . . , xd , it follows that (1.1) | Autc (G)| ≤ d Y |xG i |, i=1 since there are no more choices for the generators to go under any class-preserving automorphism. Notice that (1.1) holds true for any minimal generating set {x1 , x2 , . . . , xd } for G. Since |xG | = |[x, G]| ≤ |γ2 (G)|, from (1.1) we get (1.2) | Autc (G)| ≤ |γ2 (G)|d , where [x, G] denotes the set {[x, g] | g ∈ G} and γ2 (G) denotes the commutator subgroup of G. We say that a finite group G, minimally generated by d elements, satisfies Hypothesis A if equality holds for it in (1.2). Most obvious examples of groups satisfying Hypothesis A are abelian groups, and little less obvious ones being finite extraspecial p-groups. Notice that none of these classes of groups admit any class preserving outer automorphism. An interesting class of groups G satisfying Hypothesis A was constructed by Burnside [6] (in 1913) while answering his own question [5, page 463] about the existence of a finite group admitting a non-inner class-preserving automorphism. This group is of order p6 and is isomorphic 2010 Mathematics Subject Classification. Primary 20D15, 20D45. Key words and phrases. Camina-type group, class-preserving automorphism, p-group. 1 CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 2 to a group consisting of all 3 × 3 unitriangular matrices over the field Fp2 of p2 elements, where p is an odd prime. For this group G, Inn(G) < Autc (G) and Autc (G) is an elementary abelian p-group of order p8 . Notice that G is minimally generated by 4 elements and |γ2 (G)| = p2 . Thus it follows that | Autc (G)| = p8 = |γ2 (G)|4 , and therefore equality holds in (1.2) for G. Interestingly, a generalization of the group constructed by Burnside also enjoys this property. It follows from [2, Theorem B] that the group G consisting of all 3 × 3 unitriangular matrices over a finite field Fpm of pm elements satisfies Hypothesis A, where m ≥ 2 and p is an odd prime. A wider class of groups G satisfying Hypothesis A is the class of finite Camina p-groups of nilpotency class 2. A non-abelian finite group G is called Camina group if xG = xγ2 (G) (or equivalently [x, G] = γ2 (G)) for all x ∈ G − γ2 (G) (concept initiated by Alan Camina [7]). So, coming back to our discussion, let G be a finite Camina p-group of nilpotency class 2. Then it follows from [19, Theorem 5.2] that equality holds in (1.2). Examples of groups of larger nilpotency class and satisfying Hypothesis A are given below. A (bit wild) natural problem which arises here [21, Problem 6.7] is the following: Problem. Classify all finite p-groups G satisfying Hypothesis A. Since equality holds in (1.2) for all finite abelian groups, we only consider non-abelian ones. In [19] we considered a special case of this problem and classified, upto isoclinism (see Section 3 for the definition), all finite p-groups G such that  2 p (n 4−4) , if n is even; | Autc (G)| = 2 p (n 4−1) , if n is odd, where n = logp |G|. In this paper, continuing our work of [19], we make a substantial progress on this problem. Other motivation of this study is to provide a classification of finite nilpotent groups G having central quotient G/ Z(G) of maximum possible order. It is done in Section 11, where some historical remarks are also made on the relationship between the orders of G/ Z(G) and γ2 (G) for an arbitrary group G. Let G be a finite p-group minimally generated by d elements which satisfies Hypothesis A. Let {x1 , x2 , . . . , xd } be any minimal generating set for G. Then it is easy to see the following two statements: (i) [xi , G] = γ2 (G) for all xi , and therefore equality holds in (1.1); (ii) For any element x ∈ G − Φ(G), [x, G] = γ2 (G), where Φ(G) denotes the Frattini subgroup of G. By statement (i) it follows that equality holds in (1.1) for any minimal generating set {x1 , x2 , . . . , xd } of a finite p-group G satisfying Hypothesis A. Interestingly, the converse of this statement also holds true. We record it in the following theorem, which we prove in Section 4. Theorem A. Let G be a finite p-group. Then equality holds in (1.1) for all minimal generating sets {x1 , . . . , xd } of G if and only if equality holds for G in (1.2). The condition ‘[x, G] = γ2 (G) for all x ∈ G − Φ(G)’ (noticed in the statement (ii) above) looks like the one in the definition of a Camina group. Notice that all Camina groups enjoy this property, but the converse is not always true as shown by metacyclic p-groups, where p is an odd prime. For this reason, a non-abelian finite group G will be called Camina-type if [x, G] = γ2 (G) (or equivalently xG = xγ2 (G)) for all x ∈ G − Φ(G). Notice that a Camina-type group G is a CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 3 Camina group if and only if γ2 (G) = Φ(G) (Since the nilpotency class of finite Camina p-groups is bounded above by 3 [10], it follows from [14, Corollary 2.3] that γ2 (G) = Φ(G) for all finite Camina p-groups G). Notice (statement (ii) above) that any finite p-group G satisfying Hypothesis A is a Caminatype group. But when the nilpotency class of G is 2, then the converse also holds true (see Proposition 5.9). Thus in the case of class 2, the class of Camina-type finite p-groups coincides with the class of finite p-groups satisfying Hypothesis A. Since all finite Camina p-groups of nilpotency class 2 satisfy Hypothesis A and classification of such groups is not known (to the best of our knowledge), it seems that a complete classification of finite p-groups of class 2 satisfying Hypothesis A is a difficult task. But we have been able to obtain some interesting structural information (similar to Camina groups) for these groups, shown in the following result proved in Section 5. By d(G) we denote the number of elements in a minimal generating set of a finite group G. Theorem B. Let G be a Camina-type finite p-group of nilpotency class 2. Then the following statements hold true: (i) d(G) is even and d(G) ≥ 2d(γ2 (G)). (ii) G/ Z(G) is homocyclic. (iii) If γ2 (G) is cyclic, then G is isoclinic to a (full) central product of 2-generator groups of nilpotency class 2. Our next result is a record of the information obtained about Camina-type finite p-groups of nilpotency class larger than 2. Theorem C. Let G be a Camina-type finite p-group having nilpotency class at least 3. Then the following statements hold true: (i) d(G) = 2d(γ2 (G)/γ3 (G)) is even, where γ3 (G) denotes the third term in the lower central series of G. (ii) If γ2 (G) is cyclic, then d(G) = 2. (iii) If d(G) = 2 and |γ2 (G)/γ3 (G)| > 2, then γ2 (G) is cyclic and G satisfies Hypothesis A. We are now ready to provide a classification of finite p-groups satisfying Hypothesis A and having nilpotency class at least 3. We actually encounter some nice surprises in this case. For example, such groups can be generated by two elements. Before proceeding further, we take a little diversion to demonstrate examples of such groups (as promised above). Consider the metacylic groups (1.3) E D r+t r r+s t K := x, y | xp = 1, y p = xp , [x, y] = xp , where 1 ≤ t < r and 0 ≤ s ≤ t (t ≥ 2 if p = 2) are non-negative integers. Notice that the nilpotency class of K is at least 3. Since K is generated by 2 elements, it follows from (1.2) that | Autc (K)| ≤ |γ2 (K)|2 = p2r . It is not so difficult to see that | Inn(K)| = |K/ Z(K)| = p2r . Since Inn(K) ≤ Autc (K), it follows that | Autc (K)| = | Inn(K)| = |γ2 (K)|2 = |γ2 (K)|d(K) (That Autc (G) = Inn(G), is, in fact, true for all finite metacylic p-groups). Thus K satisfies Hypothesis A. Furthermore, if H is any 2-generator group isoclinic to K, then it follows that H satisfies Hypothesis A. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 4 After this much diversion, we come back to the main stream and provide the desired classification in the following result, which is the main theorem of this paper. Theorem D. Let G be a finite p-group of nilpotency class at least 3. Then the following statements hold true: (i) If G satisfies Hypothesis A, then d(G) = 2. (ii) Let |γ2 (G)/γ3 (G)| > 2. Then G satisfies Hypothesis A if and only if G is a 2-generator group with cyclic commutator subgroup. Moreover, G is isomorphic to some group defined in (7.2) and is isoclinic to the group K defined in (1.3) for suitable parameters. (iii) Let |γ2 (G)/γ3 (G)| = 2. Then G satisfies Hypothesis A if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian γ2 (G) of order at most 8. A detailed analysis of finite 2-generator 2-groups G satisfying Hypothesis A and with |γ2 (G)/ γ3 (G)| = 2 will be presented in [16]. Organization of the paper is as follows. Notations are set in Section 2, and some preliminary results are given in Section 3. Section 4 is devoted to proving Theorem A and some key results. Camina-type p-groups of nilpotency class 2 are discussed in Section 5, where Theorem B is proved. Section 6 deals with Camina-type p-groups with cyclic commutator subgroup. 2-generator Camina-type p-groups are studied in Section 7. 2-generator 2-groups satisfying Hypothesis A are discussed in Section 8. Macdonald’s arguments [14] are revisited in Lie theoretic setting in Section 9, which are then used to prove Theorems C and D in Section 10. As an application of Theorems B and D, a characterisation of finite nilpotent groups G such that |G/ Z(G)| = |γ2 (G)|d(G) is given in Section 11. 2. Notation Our notation for objects associated with a finite multiplicative group G is mostly standard. We use 1 to denote both the identity element of G and the trivial subgroup {1} of G. By Aut(G), Autc (G) and Inn(G), we denote the group of all automorphisms, the group of conjugacy class preserving automorphisms and the group of inner automorphisms of G respectively. The abelian group of all homomorphisms from an abelian group H to an abelian group K is denoted by Hom(H, K). We write hxi for the cyclic subgroup of G generated by a given element x ∈ G. To say that some H is a subset or a subgroup of G we write H ⊆ G or H ≤ G respectively. To indicate, in addition, that H is properly contained in G, we write H ⊂ G, H < G respectively. For a k k subgroup H of G, by H p we denote the subgroup of H generated by the set {hp | h ∈ H}, where p is a prime and k is a positive integer. If x, y ∈ G, then xy denotes the conjugate element y −1 xy ∈ G and [x, y] = [x, y]G denotes the commutator x−1 y −1 xy = x−1 xy ∈ G. If x ∈ G, then xG denotes the G-conjugacy class of all xg , for g ∈ G, and [x, G] denotes the set {[x, g] | g ∈ G} (and not the subgroup generated by these commutators). If [x, G] ⊆ Z(G), then [x, G] becomes a subgroup of G. Since xg = x[x, g], for all g ∈ G, we have xG = x[x, G]. Thus |xG | = |[x, G]|. Exponent of a subgroup H of G is denoted by exp(H). For a subgroup H of G, CG (H) denotes the centralizer of H in G and for an element x ∈ G, CG (x) denotes the centralizer of x in G. For a finitely generated group G, by d(G) we denote the number of elements in any CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 5 minimal generating set of G. By Z(G), γ2 (G) and Φ(G), we denote the center, the commutator subgroup and the Frattini subgroup of G respectively. We write the subgroups in the lower central series of G as γn (G), where n runs over all strictly positive integers. They are defined inductively by γ1 (G) = G and γn+1 (G) = [γn (G), G] for any integer n ≥ 1. Note that γ2 (G) is the commutator subgroup [G, G] of G. Let x1 , x2 , . . . , xk be k elements of G, where k ≥ 2. The commutator of x1 and x2 has been defined −1 to be [x1 , x2 ] = x−1 1 x2 x1 x2 . Now we define a higher commutator of x1 , x2 , . . . , xk inductively as [x1 , x2 , . . . , xk ] = [[x1 , . . . , xk−1 ], xk ]. If we are given a group G with a minimal generating set X, then [x1 , x2 , . . . , xk ] is said to be a commutator of weight k provided all xi ∈ X, 1 ≤ i ≤ k. 3. Preliminary results We start with the following concept of isoclinism of groups, introduced by P. Hall [11]. Let X be a group and X̄ = X/ Z(X). Then commutation in X gives a well defined map aX : X̄ × X̄ 7→ γ2 (X) such that aX (x Z(X), y Z(X)) = [x, y] for (x, y) ∈ X × X. Two groups G and H are called isoclinic if there exists an isomorphism φ of the factor group Ḡ = G/ Z(G) onto H̄ = H/ Z(H), and an isomorphism θ of the subgroup γ2 (G) onto γ2 (H) such that the following diagram is commutative a Ḡ × Ḡ −−−G−→ γ2 (G)     φ×φy yθ a H̄ × H̄ −−−H−→ γ2 (H). The resulting pair (φ, θ) is called an isoclinism of G onto H. Notice that isoclinism is an equivalence relation among groups. Let G be a finite p-group. Then it follows from [11] that there exists a finite p-group H in the isoclinism family of G such that Z(H) ≤ γ2 (H). Such a group H is called a stem group in the isoclinism family of G. The following theorem shows that the group of class-preserving automorphisms is independent of the choice of a group in a given isoclinism family of groups. Theorem 3.1 ([20], Theorem 4.1). Let G and H be two finite non-abelian isoclinic groups. Then Autc (G) ∼ = Autc (H). Lemma 3.2. Let G be a finite p-group satisfying Hypothesis A, and H be a stem group in the isoclinism family of G. Then d(G) = d(H) and H satisfies Hypothesis A. Proof. Since G satisfies Hypothesis A, it follows that Z(G) ≤ Φ(G). Also, H being a stem group, Z(H) ≤ Φ(H). Since G/ Z(G) ∼ = H/ Z(H), d(G) = d(G/ Z(G)) and d(H) = d(H/ Z(H)), we get d(G) = d(H). Now by Theorem 3.1 we have Autc (G) ∼ = Autc (H). As G satisfies Hypothesis d(G) d(H) A, | Autc (G)| = |γ2 (G)| . Thus | Autc (H)| = |γ2 (H)| , since γ2 (G) ∼ = γ2 (H). This proves that H satisfies Hypothesis A and the proof of the lemma is complete.  CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 6 The following lemma is a kind of variant of the preceding lemma with a weaker hypothesis. Lemma 3.3. Let G be a Camina-type finite p-group, and H be a stem group in the isoclinism family of G. Then d(G) = d(H) and H is also Camina-type. Proof. Let G be a group as in the statement. Then it follows that Z(G) ≤ Φ(G). Also, H being a stem group, Z(H) ≤ Φ(H). Since G/ Z(G) ∼ = H/ Z(H), d(G) = d(G/ Z(G)) and d(H) = d(H/ Z(H)), we get d(G) = d(H). Let (φ, θ) be an isoclinism of G onto H. Let h ∈ H − Φ(H) be an arbitrary element. Then there exists an element g ∈ G− Φ(G) such that φ(g Z(G)) = h Z(H). Thus by the given hypothesis [g, G] = γ2 (G). Hence θ([g, G]) = θ(γ2 (G)) = γ2 (H). By the definition of isoclinism, for any x ∈ G, θ([g, x]) = [h, y], where φ(x Z(G)) = y Z(H). Now θ([g, G]) = {θ([g, x]) | x ∈ G}. Thus γ2 (H) = θ([g, G]) = {[h, y] | y ∈ H} = [h, H], since [h, y] = 1 if y ∈ Z(H). This shows that H is a Camina-type group and the proof is complete.  The following result follows from the definition of Camina-type groups. Lemma 3.4. Let G be a finite nilpotent Camina-type group. Then G is a p-group for some prime p. The following result is due to Brady, Bryce and Cossey. Theorem 3.5 ([4], Theorem 2.1). Let G be a finite p-group of nilpotency class 2 with cyclic center. Then G is a central product either of two generator subgroups with cyclic center or two generator subgroups with cyclic center and a cyclic subgroup. The following result follows from Theorem 3.2 of Macdonald’s paper [14]. Theorem 3.6. Let G be a finite Camina p-group of class 2 such that d(G) = m and d(γ2 (G)) = n. Then m is even and 2n ≤ m. Theorem 3.7 ([19], Theorem 5.1). Let G be a finite p-group such that |G| = pn and |γ2 (G)| = pm . If | Autc (G)| = pm(n−m) , then G is a Camina group of class 2. An automorphism φ of a group G is called central if g −1 φ(g) ∈ Z(G) for all g ∈ G. The set of all central automorphisms of G, denoted by Autcent(G), is a normal subgroup of Aut(G). Notice that Autcent(G) = CAut(G) (Inn(G)). Let α ∈ Autcent(G). Then the map fα from G into Z(G) defined by fα (x) = x−1 α(x) is a homomorphism which sends γ2 (G) to 1. Thus fα induces a homomorphism from G/γ2 (G) into Z(G). So we get a one-to-one map α → fα from Autcent(G) into Hom(G/γ2 (G), Z(G)). Conversely, if f ∈ Hom(G/γ2 (G), Z(G)), then αf such that αf (x) = xf (x̄) defines an endomorphism of G, where x̄ = xγ2 (G) . But this, in general, may not be an automorphism of G. More precisely, αf fails to be an automorphism of G when G admits a non-trivial abelian direct factor. A group G is called purely non-abelian if it does not have a non-trivial abelian direct factor. The following theorem of Adney and Yen [1] shows that if G is a purely non-abelian finite group, then the mapping α → fα from Autcent(G) into Hom(G/γ2 (G), Z(G)), defined above, is also onto. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 7 Theorem 3.8 ([1], Theorem 1). Let G be a purely non-abelian finite group. Then the correspondence α → fα defined above is a one-to-one mapping of Autcent(G) onto Hom(G/γ2 (G), Z(G)). The following lemma is due to Cheng [8, Lemma 1]. Lemma 3.9. Let G = hx, yi be a 2-generator finite p-group with cyclic commutator subgroup γ2 (G) = hui of order pr for some positive integer r. Assume that either p is odd, or p = 2 and [u, G] ⊆ u4 . Then the following statements hold true: (i) γ2 (G) = {[x, y k ] | 0 ≤ k < pr }; (ii) If φ is an automorphism of G which induces the identity automorphism on G/γ2 (G), then φ ∈ Inn(G). 4. Proof of Theorem A and key results We start with the proof of Theorem A. Theorem 4.1 (Theorem A). Let G be a finite p-group. Then equality holds in (1.1) for all minimal generating sets {x1 , . . . , xd } of G if and only if equality holds for G in (1.2). Proof. Assume that | Autc (G)| = Πdi=1 |xG i | for any minimal generating set {x1 , . . . , xd } of G. G G Let y1 ∈ G − Φ(G) such that |y1 | ≥ |y | for all y ∈ G − Φ(G). Let H0 = Φ(G) and define H1 = hΦ(G), y1 i. Now inductively define Hi = hHi−1 , yi i such that |yiG | ≥ |y G | for all y ∈ G − Hi−1 , where 1 ≤ i ≤ d. Notice that the elements y1 , . . . , yd form a minimal generating set for G. So by the given hypothesis | Autc (G)| = Πdi=1 |yiG |. This implies that for given g1 , . . . , gd ∈ G, there exists an automorphism α ∈ Autc (G) such that α(yi ) = yigi , where 1 ≤ i ≤ d. First we prove that | Autc (G)| = (|y1G |)d . Since yi ∈ Hi − Hi−1 and yi−1 ∈ Hi−1 , it follows that yi−1 yi ∈ Hi − Hi−1 , where 2 ≤ i ≤ d. By our choice of yi we have |(yi−1 yi )G | ≤ |yiG |. Now given any two elements gi−1 , gi ∈ G, there exists an automorphism α ∈ Autc (G) such that g i−1 and α(yi ) = yigi . So α(yj ) = yj for all 1 ≤ j ≤ i − 2 and i + 1 ≤ j ≤ d, α(yi−1 ) = yi−1 g i−1 gi yi = α(yi−1 )α(yi ) = α(yi−1 yi ) ∈ (yi−1 yi )G . yi−1 G G This simply implies that yi−1 yiG ⊆ (yi−1 yi )G . Since (yi−1 yi )G is always contained in yi−1 yiG , we have (4.1) G yi−1 yiG = (yi−1 yi )G . G yiG | ≤ |yiG |. Since yi ∈ G − Hi−1 ⊆ This, using the fact that |(yi−1 yi )G | ≤ |yiG |, gives |yi−1 G G G − Hi−2 , the definition of yi−1 tells us that |yi | ≤ |yi−1 |. Hence G G |yiG | ≤ |yi−1 | ≤ |yi−1 yiG | ≤ |yiG |, G so that |yi−1 | = |yiG | for each i such that 2 ≤ i ≤ d. This proves that |y1G | = |y2G | = · · · = |ydG | and therefore (4.2) | Autc (G)| = (|y1G |)d . Let x be an arbitrary element of G − Φ(G). We can extend x to a minimal generating set {x1 = x, x2 , . . . , xd } of G. By the given hypothesis | Autc (G)| = Πdi=1 |xG i | and therefore G G G d G G d Πi=1 |xi | = (|y1 |) . Since |xi | ≤ |y1 | by the definition of y1 , this implies that |xG i | = |y1 |, for G G G 1 ≤ i ≤ d. Thus |xG | = |xG 1 | = |y1 |. This proves that |x | = |y | for all x, y ∈ G − Φ(G). CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 8 Now let x, y be two distinct elements of some minimal generating set for G. Then it follows from (4.1) that xG y G = (xy)G . Notice that xG = x[x, G], y G = y[y, G] and (xy)G = xy[xy, G]. The equation xG y G = (xy)G then implies that x[x, G]y[y, G] = xy[x, G]y [y, G] is equal to xy[xy, G], i. e., that (4.3) [x, G]y [y, G] = [xy, G]. Since x, y and xy all lie in G − Φ(G), it follows that |xG | = |y G | = |(xy)G |, and hence that |[x, G]| = |[y, G]| = |[xy, G]|. Since both [x, G] and [y, G] contain 1, and G is a finite group, this and (4.3) imply that (4.4) [x, G]y = [y, G] = [xy, G]. Substituting (4.4) in (4.3), we obtain [y, G][y, G] = [y, G]. Since [y, G] is a non-empty subset of the finite group G, this forces it to be a subgroup of G. Furthermore, this subgroup remains the same when y is replaced by xy, i. e., [y, G] = [xy, G]. Interchanging the role of x and y we get [x, G] = [yx, G]. Since xy and yx are conjugate, it follows that [xy, G] = [yx, G] and therefore [x, G] = [y, G]. This implies that [x1 , G] = · · · = [xd , G] = [x, G] (say) for any minimal generating set X = {x1 , . . . , xd } of G. Now we claim that [x, G] = [u, G] for every u ∈ G − Φ(G). So let u ∈ G − Φ(G) be an arbitrary element. Then u = xai11 xai22 · · · xaitt , where xij ∈ X and aj a is a positive integer for 1 ≤ j ≤ t. Notice that [xijj , G] ≤ [x, G] for all j such that 1 ≤ j ≤ t a a a and [xijj , G] = [x, G] if xijj 6∈ Φ(G). This implies that [xijj xaikk , G] ≤ [x, G], for 1 ≤ j, k ≤ t. a So [u, G] ≤ [x, G]. Since u ∈ G − Φ(G), some xijj must be outside Φ(G). This proves that [u, G] = [x, G] and hence our claim follows. So we have proved that the subgroup [y, G] is independent of the choice of y ∈ G − Φ(G) whenever d ≥ 2, i. e., whenever G is non-cyclic. Of course [y, G] = 1 is also independent of the choice of y when G is cyclic. Since the derived group γ2 (G) = [G, G] is generated by all the [y, G] for y ∈ G − Φ(G), this proves [y, G] = γ2 (G) for all y ∈ G − Φ(G). So in particular |y1G | = |[y1 , G]| = |γ2 (G)|. Hence it follows from (4.2) that | Autc (G)| = |γ2 (G)|d . Conversely suppose that | Autc (G)| = |γ2 (G)|d and {x1 , . . . , xd } is an arbitrary minimal generating set for G. Then d |γ2 (G)|d = | Autc (G)| ≤ Πdi=1 |xG i | ≤ |γ2 (G)| , d G since |xG i | ≤ |γ2 (G)| for all i between 1 and d. This proves that | Autc (G)| = Πi=1 |xi | for every minimal generating set {x1 , . . . , xd } of G. This completes the proof of the theorem.  Remark 4.2. Theorem 4.1 also holds true if we replace Hypothesis A by the fact that | Cb(G)| = |γ2 (G)|d(G) , where Cb(G) denotes the group of all basis conjugating automorphisms of G. These are the automorphisms which map each element of G − Φ(G) to its conjugate. Proof goes word to word with the proof of Theorem 4.1. The following is a key lemma, which tells that Hypothesis A passes to factor groups. Lemma 4.3. Let G be a finite p-group which satisfies Hypothesis A, and N be any non-trivial proper normal subsubgroup of G. Then G/N satisfies Hypothesis A. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 9 Proof. Let S := {x1 , x2 , . . . , xd } be a minimal generating set for G, where d is some positive integer. Since G satisfies Hypothesis A, we have | Autc (G)| = |γ2 (G)|d . Thus it follows that given any v1 , v2 , . . . , vd ∈ γ2 (G), there exists an α ∈ Autc (G) such that α(xi ) = xi vi , where 1 ≤ i ≤ d. Let N be a normal subgroup of G. Let us reorder the given generating set such that {x1 N, x2 N, . . . , xt N } generates G/N for some positive integer t ≤ d and xi N = N for all i > t. Since γ2 (G/N ) = γ2 (G)N/N , any element of γ2 (G/N ) will be of the form uN for some u ∈ γ2 (G). To establish the proof of our lemma, we only need to show that | Autc (G/N )| = |γ2 (G/N )|t . It simply requires to show that given any t elements u1 N, u2 N, . . . , ut N in γ2 (G/N ), there exists an automorphism ᾱ ∈ Autc (G/N ) such that ᾱ(xi N ) = xi N ui N for 1 ≤ i ≤ t. Let u1 , u2 , . . . , ut , ut+1 , . . . , ud ∈ γ2 (G) such that ut+1 = · · · = ud = 1. Then there exists an automorphism α ∈ Autc (G) such that α(xi ) = xi ui , where 1 ≤ i ≤ d. Since N is normal in G, α keeps N invariant and therefore induces an automorphism ᾱ (say) on G/N . Notice that ᾱ is the required automorphism of G/N , which completes the proof of the lemma.  One other key lemma tells us that like Camina groups, the property of being Camina-type group also passes to its factor groups by normal subgroups contained in the commutator subgroup. Lemma 4.4. Let G be a finite Camina-type group, and N be its normal subgroup such that N ≤ γ2 (G). Then G/N is also a Camina-type group. Proof. Let G and N be as in the statement. Consider the natural homomorphism π : G → Ḡ = G/N from G onto G/N . Let x̄ = xN ∈ Ḡ − Φ(Ḡ) be an arbitrary element. Then x ∈ G − Φ(G), and therefore [x, G] = γ2 (G). Now π(γ2 (G)) = γ2 (G)/N = γ2 (G/N ). Also π([x, G]) = [x̄, Ḡ]. Hence [x̄, Ḡ] = γ2 (G/N ), and the proof is complete.  5. Camina-type groups of class 2 In this section we study Camina-type finite p-groups of nilpotency class 2 and prove Theorem B. We start with the following result. Theorem 5.1. Let G be a Camina-type finite p-group of nilpotency class 2. Then d(G) is even and 2d(γ2 (G)) ≤ d(G). Proof. Let G be a group as given in the statement. Consider the quotient group Ḡ = G/Φ(γ2 (G)). Notice that d(G) = d(Ḡ) and d(γ2 (G)) = d(γ2 (Ḡ)). Let H be a stem group in the isoclinism family of Ḡ. Then γ2 (H) = Z(H) is of exponent p and H is a Camina-type group. Hence H is a Camina group. Now it follows from Theorem 3.6 that d(H) is even and 2d(γ2 (H)) ≤ d(H). Since d(Ḡ) = d(H) (by Lemma 3.3) and γ2 (Ḡ) ∼  = γ2 (H), assertion of the theorem follows. Let G be a finite nilpotent group of class 2. Let φ ∈ Autc (G). Then the map g 7→ g −1 φ(g) is a homomorphism of G into γ2 (G). This homomorphism sends Z(G) to 1. So it induces a homomorphism fφ : G/Z(G) → γ2 (G), sending gZ(G) to g −1 φ(g), for any g ∈ G. It is easily seen that the map φ 7→ fφ is a monomorphism of the group Autc (G) into Hom(G/Z(G), γ2 (G)). CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 10 Any φ ∈ Autc (G) sends any g ∈ G to some φ(g) ∈ g G . Then fφ (gZ(G)) = g −1 φ(g) lies in g −1 G g = [g, G]. Denote { f ∈ Hom(G/Z(G), γ2 (G)) | f (gZ(G)) ∈ [g, G], for all g ∈ G } by Homc (G/Z(G), γ2 (G)). Then fφ ∈ Homc (G/Z(G), γ2 (G)) for all φ ∈ Autc (G). On the other hand, if f ∈ Homc (G/Z(G), γ2 (G)), then the map sending any g ∈ G to gf (gZ(G)) is an automorphism φ ∈ Autc (G) such that fφ = f . Thus we have Proposition 5.2. Let G be a finite nilpotent group of class 2. Then the above map φ 7→ fφ is an isomorphism of the group Autc (G) onto Homc (G/Z(G), γ2 (G)). The following two lemmas are well known. Lemma 5.3. Let A, B and C be finite abelian groups. Then (i) Hom(A × B, C) ∼ = Hom(A, C) × Hom(B, C); (ii) Hom(A, B × C) ∼ = Hom(A, B) × Hom(A, C). Lemma 5.4. Let Cn and Cm be two cyclic groups of order n and m respectively. Then Hom(Cn , Cm ) ∼ = Cd , where d is the greatest common divisor of n and m, and Cd is the cyclic group of order d. Proposition 5.5. Let G be a finite p-group of class 2 satisfying Hypothesis A. Then G/ Z(G) is homocyclic. Proof. Let G/ Z(G) ∼ = Cpm1 × · · · × Cpmr for some integers m1 ≥ m2 ≥ · · · ≥ mr ≥ 1. Let the exponent of γ2 (G) be pe . Then the exponent of G/ Z(G) is also pe . Notice that m1 = m2 = e. We want to prove that mi = e for each i such that 1 ≤ i ≤ r. If r = 2, we are done. So suppose that r ≥ 3. Contrarily assume that mk < e for some k such that 3 ≤ k ≤ r. Now (5.1) Hom(G/ Z(G), γ2 (G)) ∼ = ∼ = Hom(Cpm1 × · · · × Cpmr , γ2 (G)) Hom(Cpm1 , γ2 (G)) × · · · × Hom(Cpmr , γ2 (G)). By considering cyclic factorization of γ2 (G), it is easy to show that | Hom(Cpmk , γ2 (G))| < |γ2 (G)|. Now using (5.1), we have | Autc (G)| = |Homc (G/Z(G), γ2 (G))| ≤ |Hom(G/Z(G), γ2 (G))| < |γ2 (G)|r . Notice that Z(G) ≤ Φ(G). Thus r = d(G) and therefore | Autc (G)| < |γ2 (G)|d(G) . This contradicts the fact that G satisfies Hypothesis A. Hence mi = e for each i such that 1 ≤ i ≤ r, which proves that G/ Z(G) is homocyclic.  We can even prove the above proposition directly for Camina-type p-groups using different methods. Proposition 5.6. Let G be a Camina-type finite p-group of class 2. Then G/ Z(G) is homocyclic. Proof. Since G is a Camina-type finite p-group of class 2, Z(G) ≤ Φ(G). Let x1 , . . . , xd be a minimal generating set for G such that G/ Z(G) = hx1 Z(G)i × · · · × hxd Z(G)i be a cyclic (direct) decomposition of G/ Z(G). Notice that such a minimal generating set always exists for CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 11 any finite p-group K of class 2 with Z(K) ≤ Φ(K) ([22, Lemma 3.5(1)]). Let the exponent of γ2 (G) be pe for some positive integer e and the order of xi modulo Z(G) be pei for 1 ≤ i ≤ d. Since pei ≤ pe = exp(G/ Z(G)), to complete the proof, it is sufficient to show that pei ≥ pe for each i. Let u ∈ γ2 (G) be such that order of u is pe . Since G is a Camina-type group, [xi , G] = γ2 (G) for 1 ≤ i ≤ d. Thus, for each i, there exists some gi ∈ G such that u = [xi , gi ]. ei ei ei ei Now up = [xi , gi ]p = [xpi , gi ] = 1, since xpi ∈ Z(G). This shows that pe ≤ pei for each i such that 1 ≤ i ≤ d and the proof of the lemma is complete.  Next two results are about the relationship between class-preserving and central automorphisms. Lemma 5.7. Let G be a Camina-type finite p-group of class 2, and H be a stem group in the isoclinism family of G. Then Autcent(H) = Autc (H). Proof. Let G and H be the groups as in the statement. Since G is a Camina-type p-group, Lemma 3.3 tells that H is also a Camina-type p-group. Since Z(H) = γ2 (H), H is purely non-abelian. It now follows from Theorem 3.8 that there is a bijection between Autcent(H) and Hom(H/γ2 (H), Z(H)). Since Autc (H) ≤ Autcent(H) for any p-group H of nilpotency class 2, we only need to show that Autcent(H) ≤ Autc (H) to complete the proof. Let α ∈ Autcent(H) be an arbitrary central automorphism of H. Then fα ∈ Hom(H/γ2 (H), Z(H)). Since H is a Caminatype group, for any x ∈ H − Φ(H) we have [x, H] = γ2 (H) = Z(H). Thus fα (xγ2 (H)) ∈ [x, H]. Let xγ2 (H) ∈ Φ(H)/γ2 (H). Then there exists an element y ∈ H − Φ(H) such that x = y t (modulo γ2 (H)) for some integer t. Notice that [x, H] = [y t , H] = [y, H]t , since γ2 (H) = Z(H). Thus fα (xγ2 (H)) = fα (y t γ2 (H)) = fα (yγ2 (H))t ∈ [y, H]t = [x, H]. Hence fα ∈ Homc (H/ Z(H), γ2 (H)) and therefore α ∈ Autc (H) (by Proposition 5.2). This proves that Autcent(H) ≤ Autc (H).  In [22] we proved that if Autcent(G) = Autc (G) for any finite p-group G, then Z(G) = γ2 (G). Using this and the above lemma, we here get Proposition 5.8. Let G be a Camina-type finite p-group of class 2. Then Autcent(G) = Autc (G) if and only if Z(G) = γ2 (G). As we noticed in the introduction that any finite p-group satisfying Hypothesis A is a Caminatype group. We now prove the converse of this statement in the case of nilpotency class 2. Proposition 5.9. Let G be a Camina-type finite p-group of nilpotency class 2. Then G satisfies Hypothesis A. Proof. Suppose that H is a stem group in the isoclinism family of G. Then, H being Caminatype (by Lemma 3.2), by Proposition 5.6 and Lemma 5.7, we have H/ Z(H) is homocyclic and Autcent(H) = Autc (H) respectively. Hence | Autc (H)| = | Autcent(H)| = Hom(H/γ2 (H), Z(H)) = Hom(H/ Z(H), γ2 (H)) = |γ2 (H)|d(H) , CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 12 since γ2 (H) = Z(H). This proves that H satisfies Hypothesis A. That G satisfies Hypothesis A, now can be shown on the lines of the proof of Lemma 3.2, since Z(G) ≤ Φ(G). This completes the proof of the proposition.  Now onwards, in this section, we mainly concentrate on characterizing finite p-groups G of class 2 satisfying Hypothesis A and such that Autc (G) = Inn(G). We first prove Theorem 5.10. Let G be a finite p-group of class 2 satisfying Hypothesis A. Then Autc (G) = Inn(G) if and only if γ2 (G) is cyclic. Proof. Since Autc (G) = Inn(G) for any finite p-group G of nilpotency class 2 with γ2 (G) cyclic ([20, Corollary 3.6]), if part of the assertion follows. Now suppose that Autc (G) = Inn(G). Let H be a stem group in the isoclinism family of G. Since γ2 (G) ∼ = γ2 (H), it is sufficient to prove that γ2 (H) is cyclic. Notice that Autc (H) = Inn(H). Also notice that H is a Camina-type p-group. Hence by Proposition 5.6 and Lemma 5.7, we have H/ Z(H) is homocyclic of exponent equal the exponent of γ2 (H) and Autcent(H) = Autc (H) respectively. Suppose contrarily that γ2 (G) ∼ = γ2 (H) is not cyclic. Then γ2 (H) ∼ = Cpe ×K, where pe is the exponent of γ2 (H) and K is some non-trivial subgroup of H contained in γ2 (H). Since γ2 (H) = Z(H), by Theorem 3.8 we have | Autcent(H)| = = | Hom(H/ Z(H), γ2 (H))| = | Hom(H/ Z(H), Cpe ) × Hom(H/ Z(H), K)| |H/ Z(H)|| Hom(H/ Z(H), K)|. This gives | Inn(H)| = | Autc (H)| = |H/ Z(H)|| Hom(H/ Z(H), K)| > | Inn(H)|, which is not possible. Hence γ2 (H) is cyclic and the proof of the theorem is complete.  Let G be a 2-generator finite p-group of class 2. Then γ2 (G) is cyclic and therefore | Autc (G)| = | Inn(G)| = |γ2 (G)|2 . Thus G satisfies Hypothesis A. Let q = pe be the order of γ2 (G). Then |G/ Z(G)| = q 2 . If we assume that γ2 (G) = Z(G), then |G| = q 3 . More generally, for any positive integer m, let G1 , G2 , . . . , Gm be 2-generator finite p-groups such that γ2 (Gi ) = Z(Gi ) ∼ = X (say) is cyclic of order q for 1 ≤ i ≤ m. Consider the central product (5.2) Y = G1 ∗X G2 ∗X · · · ∗X Gm of G1 , G2 , . . . , Gm amalgamated at X (isomorphic to cyclic commutator subgroups γ2 (Gi ), 1 ≤ i ≤ m). Then |Y | = q 2m+1 and |Y / Z(Y )| = q 2m = |γ2 (Y )|d(Y ) as d(Y ) = 2m. Since γ2 (Y ) is cyclic, Autc (Y ) = Inn(Y ) and therefore Y satisfies Hypothesis A. In the following result, we show that these are the only finite p-groups (upto isoclinism) of class 2 with cyclic commutator subgroup and satisfying Hypothesis A. Theorem 5.11. Let G be a finite p-group of class 2 satisfying Hypothesis A and with cyclic commutator subgroup. Then G is isoclinic to the group Y , defined in (5.2), for a suitable positive integer m. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 13 Proof. Let G be a group as in the statement and H be a stem group in the isoclinism family of G. Then γ2 (H) = Z(H) is cyclic of order q = pe (say) and, by Proposition 5.5, H/ Z(H) is homocyclic of exponent q and is of order q 2m for some positive integer m, since d(H) is even (by Theorem 5.1). Since Z(H) = γ2 (H) is cyclic, it follows from Theorem 3.5 that H is a central product of 2-generator groups H1 , H2 , . . . , Hm . Now it is easy to see that γ2 (Hi ) = Z(Hi ) and |γ2 (Hi )| = q for 1 ≤ i ≤ m. This completes the proof of the theorem.  We are now ready to prove Theorem B. Proof of Theorem B. Let G be a Camina-type finite p-group of nilpotency class 2. Then by Proposition 5.9 G satisfies Hypothesis A. That d(G) is even and d(G) ≥ 2d(γ2 (G)), now follows from Theorem 5.1. Proposition 5.6 tells that G/ Z(G) is homocyclic. The last assertion is Theorem 5.11.  6. Camina-type groups with cyclic commutator subgroup In this section we show that all Camina-type finite p-groups of nilpotency class ≥ 3 with cyclic commutator subgroup are two generator groups. We start with the following interesting result. Lemma 6.1. Let G be a Camina-type finite p-group of class 3 such that |γ3 (G)| = p. Then the following statements hold true in G: (6.1a) (γ2 (G))p ≤ Z(G). (6.1b) Z2 (G) = Φ(G), where Z2 (G) denotes the second center of G. Proof. Suppose that u ∈ γ2 (G) and x ∈ G. Then [u, x] ∈ γ3 (G) ≤ Z(G), and therefore [up , x] = [u, x]p = 1 as |γ3 (G)| = p. Since this holds true for all u ∈ γ2 (G) and x ∈ G, it follows that (γ2 (G))p ≤ Z(G). This proves (6.1a). Now we prove (6.1b). First we show that Z2 (G) ≤ Φ(G). Contrarily assume that Z2 (G) ≤ 6 Φ(G). Let x ∈ Z2 (G) − Φ(G). Since G is Camina-type, [x, G] = γ2 (G). Since x ∈ Z2 (G), it follows that γ2 (G) = [x, G] ≤ Z(G). This contradicts the fact that the nilpotency class of G is 3. Hence Z2 (G) ≤ Φ(G). Next suppose that Z2 (G) < Φ(G). Since γ2 (G) ≤ Z2 (G), we can always find an element x ∈ Φ(G) − Z2 (G) and an element y ∈ G − Z2 (G) such that x = y p . For, if y p ∈ Z2 (G) for all y ∈ G − Z2 (G), then Gp ≤ Z2 (G) and therefore Φ(G) ≤ Z2 (G) (since γ2 (G) ≤ Z2 (G)), which we are not considering. Now for any arbitrary element v ∈ G, we have [x, v] = [y p , v] = [y, v]p modulo Z(G). But [y, v]p ∈ Z(G) by (6.1a). Hence [x, v] ∈ Z(G) for all v ∈ G. This gives that x ∈ Z2 (G), which is a contradiction to our supposition that x ∈ Φ(G) − Z2 (G). Hence Z2 (G) = Φ(G), which proves (6.1b) and the proof of the lemma is complete. The following result is also of independent interest.  CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 14 Lemma 6.2. Let G be a finite p-group of class 3 such that γ2 (G) Z(G)/ Z(G) is cyclic. Then G/ Z2 (G) is generated by 2 elements. Proof. Let {x1 , x2 , . . . , xd } be a minimal generating set for G. Since γ2 (G) Z(G)/ Z(G) is cyclic, we can assume without loss of generality that [x1 , x2 ] Z(G) generates γ2 (G) Z(G)/ Z(G). Then [x1 , xj ] = [x1 , x2 ]αj modulo Z(G) for some integer αj , where 3 ≤ j ≤ d. Since the nilpotency −α class of G/ Z(G) is 2, it follows that [x1 , xj x2 j ] ∈ Z(G). Similarly if [x2 , xj ] = [x1 , x2 ]βj β modulo Z(G) for some integer βj , where 3 ≤ j ≤ d, then [x2 , x1 j xj ] ∈ Z(G). Thus it follows that β β −α −α [xi , x1 j xj x2 j ] ∈ Z(G) for all i, j such that 1 ≤ i ≤ 2 and 3 ≤ j ≤ d. Let us set yj = x1 j xj x2 j , where 3 ≤ j ≤ d. Notice that the set {x1 , x2 , y3 , . . . , yd } also generates G. Thus it follows that [xi , yj ] ∈ Z(G) for all 1 ≤ i ≤ 2 and 3 ≤ j ≤ d. Also [yj , xi ] ∈ Z(G) for the same values of i and j. Therefore by Hall-Witt identity [xi , yj , yk ][yk , xi , yj ][yj , yk , xi ] = 1 for 1 ≤ i ≤ 2 and 3 ≤ j, k ≤ d, we get (6.2) [yj , yk , xi ] = 1. Similarly [x1 , x2 , yl ] = 1, since [yl , x1 , x2 ] = 1 = [x2 , yl , x1 ] = 1, where 3 ≤ l ≤ d. Now for all j, k such that 3 ≤ j, k ≤ d, we have [yj , yk ] = [x1 , x2 ]t z for some integer t and some z ∈ Z(G). Since [x1 , x2 , yl ] = 1, this gives [yj , yk , yl ] = 1, where 3 ≤ j, k, l ≤ d. This, along with (6.2), proves that [yj , yk ] ∈ Z(G) for all j, k such that 3 ≤ j, k ≤ d. Since [yj , xi ] ∈ Z(G) for 3 ≤ j ≤ d and 1 ≤ i ≤ 2, it follows that [yj , G] ⊆ Z(G) for all j such that 3 ≤ j ≤ d. Hence yj ∈ Z2 (G) for 3 ≤ j ≤ d. This proves that G/ Z2 (G) is generated by x1 Z2 (G) and x2 Z2 (G), which completes the proof of the lemma.  Proposition 6.3. Let G be a Camina-type finite p-group of nilpotency class 3 such that |γ3 (G)| = p and γ2 (G) Z(G)/ Z(G) is cyclic. Then G is generated by 2 elements. Proof. Let G be a group satisfying all the conditions of the statement. Then it follows from Lemma 6.1 that (γ2 (G))p ≤ Z(G) and Z2 (G) = Φ(G). Since γ2 (G) Z(G)/ Z(G) is cyclic, it follows from Lemma 6.2 that G/ Z2 (G) is generated by 2 elements and therefore G/Φ(G) is generated by 2 elements, since Z2 (G) = Φ(G). This shows that G is generated by 2 elements and the proof of the proposition is complete.  Here is the main result of this section. Theorem 6.4. If G is a Camina-type finite p-group of nilpotency class at least 3 with cyclic commutator subgroup, then d(G) = 2. Proof. Let G be a finite Camina-type p-group of nilpotency class at least 3. Then there exists a maximal subgroup N of γ3 (G) which is normal in G. Thus it follows from Lemma 4.4 that G/N is a Camina-type group. Also G/N is of nilpotency class 3 such that γ3 (G/N ) is cyclic of order p. Thus it follows from Proposition 6.3 that G/N is generated by 2 elements. Since N ≤ Φ(G), if follows that both G as well as G/N are minimally generated by equal number of elements. Hence G is generated by 2 elements and the proof is complete.  CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 15 7. 2-generator Camina-type p-groups In this section we study finite 2-generator Camina-type p-groups. The following lemma provides a sort of converse of Theorem 6.4. Lemma 7.1. Let H be a 2-generator Camina-type finite p-group of nilpotency class at least 3 such that |γ2 (H)/γ3 (H)| > 2. Then γ2 (H) is cyclic. Proof. Let H be the group as given in the statement and M be a subgroup of γ3 (H) of index p which is normal in H. Then we can always choose a minimal generating set {x, y} of H such that x̄ ∈ CH̄ (γ2 (H̄)), where H̄ = H/M and x̄ = xM . For, let H = hx′ , yi such that neither x̄′ nor ȳ centralize γ2 (H̄). Since γ3 (H̄) ≤ Z(H̄), it gives that neither [x̄′ , ū] nor [ȳ, ū] is the identity element of H̄, where uγ3 (H) is a generator of the cyclic group γ2 (H)/γ3 (H). Since both [x̄′ , ū] and [ȳ, ū] lie in γ3 (H̄) and γ3 (H̄) is cyclic of order p, there exists an integer t, 1 ≤ t < p, such that [ȳ, ū]t = [x̄′ , ū]. Which implies that [y −t x′ , u] = 1 modulo M . Hence {x = y −t x′ , y} is a generating set for H with the required property. Notice that [x, y] = [y −t x′ , y] = [x′ , y]. For the rest of the proof, we fix such a generating set {x, y} for H and assume that u = [x, y]. Also let the orders of x and y modulo Z(H) be pm and pn respectively. For the clarity of the exposition, we complete the proof in 3 steps. Step 1. If |γ3 (H)| = p, then γ2 (H) is cyclic. Proof. Since |γ3 (H)| = p, the subgroup M considered above is 1. So the generating set {x, y} of H is such that x ∈ CH (γ2 (H)) (or equivalently γ2 (H) ≤ CH (x)). If [x, y s ] = 1 for some integer s, s then y s ∈ Z(H). Thus it follows that for no positive integer s < n, [x, y p ] = 1, since the order y t modulo Z(H) is pn . Similarly for no positive integer t < m, [xp , y] = 1. Let h ∈ H be such that [x, h] = 1. Then h = vxi y j for some v ∈ γ2 (H) and some integers i, j. Since γ2 (H) ≤ CH (x), [x, h] = 1 implies that [x, y j ] = 1. Thus j is either zero, or a power of p larger than or equal to pn , and therefore CH (x) ≤ γ2 (H) Z(H) hxi. Obviously CH (x) ≥ γ2 (H) Z(H) hxi. Hence CH (x) = γ2 (H) Z(H) hxi. Since H is a Camina-type group, it follows that [x, H] = γ2 (H). Thus |γ2 (H)| = |[x, H]| = |G : CH (x)| = pn . Since |γ3 (H)| = p, this implies that |γ2 (H)/γ3 (H)| = pn−1 . Again, since [x, H] = γ2 (H) and γ2 (H) Z(H) hxi = CH (x), there exists an element h ∈ H − γ2 (H) Z(H) hxi of the form h = y k for some positive integer k such that h[x, h]i = γ3 (H) and 1 ≤ k < pn . Without loss of generality, we can assume that k is some power of p (by modifying h, if necessary). It now follows that k = pn−1 . Notice that n ≥ 2, otherwise h = y, which is n−1 n−1 not possible as [x, y] generates γ2 (H) modulo γ3 (G). If p is odd, [x, y p ] = [x, y]p , since |γ3 (H)| = p. If p = 2, then our hypothesis |γ2 (H)/γ3 (H)| > 2 tells that pn = |γ2 (G)| ≥ 23 , n−1 n−1 n−1 n−1 which implies that n ≥ 3. Hence [x, y p ] = [x, y]p . Set w = [x, h] = [x, y p ] = [x, y]p Since γ2 (H) is generated by γ3 (H) = hwi and [x, y], it follows that γ2 (H) is cyclic. Step 2. If γ4 (H) = 1, then γ2 (H) is cyclic. . CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 16 Proof. First we show that γ3 (H) is cyclic. Contrarily assume that γ3 (H) is not cyclic. Since, by Lemma 4.4, factor group of H by any normal subgroup contained in γ2 (H) is Camina-type, it is sufficient to assume that γ3 (H) ∼ = Cp × Cp (by taking H as H/(γ3 (H))p ). Notice that γ3 (H) is generated by w1 := [[x, y], x] and w2 := [[x, y], y]. Since [x, H] = γ2 (H), there is an element h ∈ H such that w2 = [x, h]. Since γ3 (H) ≤ Z(H), h can be taken of the form y k [x, y]l for some integers k and l. Now (7.1) w2 = [x, h] = [x, y k [x, y]l ] = [x, [x, y]l ][x, y k ] = w1−l [x, y k ]. Since the order of y modulo Z(H) is pn , notice that k is not congruent to 0 modulo pn . Otherwise w2 will be a power of w1 , which is not the case. Now (7.1) tells us that 1 6= [x, y k ] ∈ γ3 (H). Hence [x, y k ]p = [x, y pk ] = 1, and therefore y pk ∈ Z(H). Then k = k1 pn−1 for some non-zero positive n−1 integer k1 coprime to p. If n = 1, then obviously [x, y k ] = [x, y k1 ] = [x, y k1 ]p . So let n > 1. Since the exponent of γ3 (H) is p, in both of the cases, i.e., p is odd or p = 2 and |γ2 (H)/γ3 (H)| > n−1 2, it follows that [x, y k ] = [x, y k1 ]p . Notice that [x, y k1 ] = [[x, y], y]k1 (k1 −1)/2 [x, y]k1 = n−1 n−1 k (k −1)/2 = [x, y]k1 p = [x, y]k , since w2 1 1 [x, y]k1 . Therefore, for n ≥ 2, [x, y k ] = [x, y k1 ]p the exponent of γ3 (H) is p. Thus (7.1) tells us that w1l w2 is a power of [x, y]. We now claim that n is always ≥ 2. For, if n = 1, then k = k1 . Thus [x, y k ] = [[x, y], y]k(k−1)/2 [x, y]k . Since [x, y k ] ∈ γ3 (H), this implies that [x, y]k ∈ γ3 (H), which is not possible because k = k1 is coprime to p. Obviously the set {w = w1l w2 , w1 } generates γ3 (H). Let N = hwi. Then N is a normal subgroup of H of order p. Set K = H/N . Then, notice that, K is a Camina-type group and γ3 (K) is cyclic of order p. Now we apply Step 1 to conclude that modulo N , w1 is a power of [x, y]. Since N is generated by a power of [x, y], it follows that w1 is a power of [x, y]. Thus each element of γ3 (H) is a power of [x, y], which is a contradiction. Hence γ3 (H) is cyclic. Let T := hwp i be the maximal subgroup of γ3 (H) := hwi generated by wp . Thus T is normal in H. Consider the quotient group H/T , which satisfies all the given hypotheses and |γ3 (H/T )| = p. Hence by Step 1 wT is a power of [x, y]T , and therefore there exist positive α2 integers α1 , α2 such that wK = [x, y]α1 p T , where α1 is coprime to p. Since T is generated by α2 wp , there exists a non-zero integer β such that w−1 [x, y]α1 p = wpβ . Hence wpβ+1 is a power of [x, y]. Since wpβ+1 also generates γ3 (H), and γ2 (H)/γ3 (H) is generated by [x, y]γ3 (H), it follows that γ2 (H) is a cyclic subgroup generated by [x, y]. Step 3. γ2 (H) is always cyclic. Proof. We are going to show that γ2 (H) is generated by [x, y]. We’ll do this by proving that γ2 (H)/γi (H) is generated by [x, y]γi (H) for 4 ≤ i ≤ c + 1, where c denotes the nilpotency class of H. That γ2 (H)/γ4 (H) is generated by [x, y]γ4 (H) follows from Step 2 as H/γ4 (H) is of nilpotency class 3. Thus the elements [x, y, x] and [x, y, y] are powers of [x, y] modulo γ4 (H). Notice that γ4 (H) is generated by γ5 (H) and the elements [x, y, x, x], [x, y, x, y], [x, y, y, x] and [x, y, y, y]. To make the reader little comfortable, we first show that γ2 (H)/γ5 (H) is cyclic. It is sufficient to prove that γ2 (S) is cyclic, where S = H/(γ4 (H)p γ5 (H)). We’ll now work in S and all elements are considered modulo γ4 (H)p γ5 (H). Notice that γ4 (S) is elementary abelian group generated by [x, y, x, x], [x, y, x, y], [x, y, y, x] and [x, y, y, y]. Let [x, y, x] = [x, y]α v1 and [x, y, y] = [x, y]β v2 , where α and β are non-zero integers which are multiples of p, and CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 17 v1 , v2 ∈ γ4 (S). Now [x, y, x, y] = [[x, y]α v1 , y] = [[x, y]α , y] = ([x, y]β v2 )α = [x, y]αβ , since v1 , v2 ∈ Z(S) and α is a multiple of p. Similarly, we can show that other generators of γ4 (S) are also some powers of [x, y]. This proves that γ2 (S) is a cyclic subgroup of S generated by [x, y] (modulo γ4 (H)p γ5 (H)). Hence it follows that γ2 (H)/γ5 (H) is cyclic. Now, by inductive argument, assume that γ2 (H)/γi (H) is a cyclic group generated by [x, y]γi (H). We prove that γ2 (H)/γi+1 (H) is cyclic. Consider S = H/(γi (H)p γi+1 (H)). Again we read elements modulo γi (H)p γi+1 (H). Notice that γi (S) is an elementary abelian central subgroup of S generated by at most 2i−2 commutators of weight i. But we know that all commutators of weight i − 1 are powers of [x, y] modulo γi (H). So a given commutator c of weight ′ ′ i in S can be written as [[x, y]α v1′ , x] or [[x, y]α v1′ , y], where α′ is a non-zero integer which is a ′′ ′′ multiple of p, and v1′ ∈ γi (S). Also [x, y, x] = [x, y]α v1′′ and [x, y, y] = [x, y]β v2′′ , where α′′ and β ′′ are non-zero integers which are multiples of p, and v1′′ , v2′′ ∈ γi (S). Since γi (S) is elementary abelian central subgroup of S, it is easy to see that c is some power of [x, y] in S. Hence it follows that γ2 (H)/γi+1 (H) is cyclic. Proof is now complete by inductive argument.  We are now going to show that most of 2-generator groups with cyclic commutator subgroup satisfy Hypothesis A, and therefore these are Camina-type groups. Let G be a 2-generator finite p-group of nilpotency class at least 3 with cyclic commutator subgroup γ2 (G) = hui. Assume that |γ2 (G)/γ3 (G)| > 2. Notice that the condition |γ2 (G)/γ3 (G)| > 2 is equivalent to the condition [u, G] ⊆ u4 . Set C = CG (γ2 (G)). Then G/C is cyclic, because it is isomorphic to the cyclic group of automorphisms of γ2 (G) induced by G by conjugation. Let y ∈ G be such that G/C = hyCi. Then there exists an element x ∈ C such that G = hx, yi and [x, u] = 1. Let B = hx, ui. Then B is an abelian normal subgroup of G and G = B hyi. Set |γ2 (G)| = pr for some positive integer r. Since the order of γ2 (G) is pr and γ2 (G) = {[x, y k ] | 0 ≤ k < pr }(= {[xk , y] | 0 ≤ k < pr }) (by Lemma 3.9), it follows that |G/ Z(G)| ≥ p2r . It again follows from ∼ Inn(G) ≤ Autc (G) = Inn(G). Since, G being minimally generated Lemma 3.9 that G/ Z(G) = by two elements, | Autc (G)| ≤ |γ2 (G)|2 = p2r . We get | Autc (G)| = |G/ Z(G)| = p2r . This shows that G satisfies Hypothesis A. Thus we have proved Lemma 7.2. Let G be a 2-generator finite p-group of nilpotency class at least 3 such that γ2 (G) is cyclic and |γ2 (G)/γ3 (G)| > 2. Then G satisfies Hypothesis A, and therefore G is Camina-type group. Combining Lemmas 7.1 and 7.2, we get Corollary 7.3. If G is a 2-generator Camina-type finite p-group such that |γ2 (G)/γ3 (G)| > 2, then γ2 (G) is cyclic and G satisfies Hypothesis A. The preceding discussion gives: CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 18 Proposition 7.4. Let G be a 2-generator finite p-group of nilpotency class at least 3 such that γ2 (G) is cyclic and |γ2 (G)/γ3 (G)| > 2. Then G can be presented by E D m n r s (7.2) G = x, y, u | xp = ui , y p = xj uk , [x, y] = u, up = 1 = [u, x], [u, y] = up , where m, n, i, j, k, s are some non-negative integers such that 1 ≤ s < r (s ≥ 2 if p = 2). Theorem 7.5. Let G be a 2-generator Camina-type finite p-group of nilpotency class at least 3 such that |γ2 (G)/γ3 (G)| > 2. Then G is isoclinic to some group defined in (1.3). Proof. Let G be a group as in the statement. Then Corollary 7.3 tells that G satisfies Hypothesis A and γ2 (G) is cyclic. Set |γ2 (G)| = pr . Let K be a stem group in the isoclinism family of G. Then Z(K) ≤ γ2 (K) and K satisfies Hypothesis A, and therefore |K/ Z(K)| = |γ2 (K)|2 . Let us assume that K is generated by x and y. Since γ2 (K) is cyclic, we can assume, without loss of any generality, that x commutes with the generator [x, y] of γ2 (K). Since [x, K] = γ2 (K) and γ2 (K) ≤ CK (x), it follows that no power of y smaller than pr can land in Z(K), and therefore no such power of y can lie in γ2 (K). Let us see what can lie in γ2 (K). Let xi y j ∈ γ2 (K). Since γ2 (K) ≤ CK (x), it follows that x commutes with xi y j . This tells that [x, y j ] = 1, and therefore y j ∈ Z(K). Hence j must be either 0 or some power of pr . So it follows that modulo Z(K), r s1 anybody from K − K p that can lie in γ2 (K) must be some power of x. Suppose that xp is the smallest power of x that lies in γ2 (K). We know for sure that this s1 ≤ r, since Z(K) ≤ γ2 (K). Set | Z(K)| = pt . Notice that |K/ Z(K)| = |K/γ2 (K)||γ2 (K)/ Z(K)| = pr+s1 pr−t = p2r+s1 −t , since |K/γ2 (K)| = pr+s1 and |γ2 (K)/ Z(K)| = pr−t . But we know that K/ Z(K) is of order p2r . Hence the preceding equation implies that s1 = t. Since the order of x modulo Z(K) is t pr , it follows that xp generates γ2 (K). This proves that K is metacyclic. Order of x is pr+t , r r+s t y p = xp and [x, y] = xp , for some non-negative integer s. If r = 1, then |γ2 (K)| = p, and therefore nilpotency class of K is 2, which we are not considering. Hence r ≥ 2. It is now clear that 1 ≤ t < r and 0 ≤ s ≤ t. Indeed, if t = r, then Z(K) = γ2 (K), which we are not considering. This completes the proof of the theorem.  As a consequence of the preceding theorem and Lemma 7.2 we get Corollary 7.6. Let G be a 2-generator finite p-group of nilpotency class at least 3 such that γ2 (G) is cyclic and |γ2 (G)/γ3 (G)| > 2. Then G is isoclinic to a metacyclic p-group. We conclude this section with the following specific non-metacylic p-group satisfying Hypothesis A. For an odd prime integer p, consider the group G = 2 2 2 hx, y, u | xp = y p = up = 1, [x, y] = u, [u, x] = up , [u, y] = 1i. It is not difficult to show that G is a 2-generator group of order p6 , the nilpotency class of G is 3, γ2 (G) and Z(G) are distinct cyclic subgroups of order p2 and γ3 (G) is of order p. Since γ2 (G) is cyclic, Autc (G) = Inn(G). Also | Inn(G)| = |G/ Z(G)| = p4 = |γ2 (G)|2 . Hence G satisfies Hypothesis A. But G is not metacyclic. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 19 8. 2-generator 2-groups G with |γ2 (G)/γ3 (G)| = 2 In this section we classify finite 2-generator 2-groups G satisfying Hypothesis A and with |γ2 (G)/γ3 (G)| = 2. These results are proved jointly with Mike Newman. It is easy to prove the following two results using computer algebra system Magma [3]. For a given group G, let Y (G) denote the normal subgroup γ4 (G)(γ2 (G))2 of G. Lemma 8.1. Let G be a finite 2-generator 2-group satisfying Hypothesis A and |γ2 (G)/γ3 (G)| = 2. If the order of G is at most 28 , then Y (G) = 1. Lemma 8.2. Let G be a finite 2-generator 2-group such that the nilpotency class of G is at most 3, order of G is at most 27 and γ2 (G) is elementary abelian of order at most 8. Then G satisfies Hypothesis A. Furthermore, Autc (G) = Inn(G) if and only if |γ2 (G)| ≤ 4. Theorem 8.3. Let G be a finite 2-generator 2-group such that |γ2 (G)/γ3 (G)| = 2. Then G satisfies Hypothesis A if and only if the nilpotency class of G is at most 3 and γ2 (G) is elementary abelian of order at most 8. Proof. Let G be a group as in the statement, which is generated by x and y. Suppose that G satisfies Hypothesis A. To prove the ‘only if’ part of the statement, it is sufficient to show that Y (G) = 1. Contrarily assume that Y (G) = 6 1. Then, G being a finite 2-group, there exists a subgroup M of index 2 in Y (G) which is normal in G. Now consider the quotient group H = G/M . Observe that |Y (H)| = 2 (for, trivial Y (H) implies Y (G) = M , which is not true), and the order and exponent of γ2 (H) are at most 16 and 4 respectively. Now it follows that both x̄4 and ȳ 4 lie in Z(H), where x̄ = xM and ȳ = yM generate H. Let K be a stem group in the isoclinism family of H. Then K is of order at most 28 and Y (K) 6= 1. Also K is 2-generator and satisfies Hypothesis A (by Lemma 3.2), and |γ2 (K)/γ3 (K)| = 2. This is not possible by Lemma 8.1. Hence Y (K) must be trivial, and therefore so is Y (G). Conversely, assume that the nilpotency class of G is at most 3 and γ2 (G) is elementary abelian of order at most 8. Let K be a stem group in the isoclinism family of G. Since both x4 and y 4 lie in Z(G), it follows that the order of K is at most 27 . Since Z(K) ≤ γ2 (K), we have d(K) = d(G) = 2. Then Lemma 8.2 tells that K satisfies Hypothesis A. Since Autc (G) ∼ = Autc (K) (by Theorem 3.1), it now follows that G satisfies Hypothesis A. This completes the proof of the theorem.  The existence of 2-generator 2-groups G satisfying Hypothesis A, having nilpotency classes 2 and 3, and having orders 32, 64, 128, 256 and even higher with |γ2 (G)/γ3 (G)| = 2 can be easily established using Magma. Using second assertion of Lemma 8.2, the following result follows from Theorem 8.3. Proposition 8.4. Let G be a finite 2-generator 2-group of nilpotency class at least 3 such that |γ2 (G)/γ3 (G)| = 2 and Autc (G) = Inn(G). Then G satisfies Hypothesis A if and only if the nilpotency class of G is 3 and γ2 (G) is elementary abelian of order 4. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 20 9. Camina-type p-groups: Macdonald’s argument We begin with some standard observations (which can be found in [13]) about an arbitrary prime p and an arbitrary finite p-group G. In this situation there is a natural graded Lie ring (9.1) L = L1 ⊕ L2 ⊕ . . . with i-th component (9.2) Li = (γi (G)/γi+1 (G))+ , for any integer i ≥ 1, is the abelian section γi (G)/γi+1 (G) of G rewritten as an additive group instead of multiplicative one. So any element xi ∈ Li is a coset xγi+1 (G) for some x ∈ γi (G). Similarly, any yj ∈ Lj , where j = 1, 2, . . ., is a coset yγj+1 (G) for some y ∈ γj (G). Then the Lie product [xi , xj ] in the Lie ring L is the coset (9.3) [xi , xj ] = [x, y]γi+j+1 (G) ∈ Li+j of the commutator [x, y] ∈ γi+j (G) of x and y in G. This product is well defined, and turns L into a Lie ring. The usual equation γi+1 (G) = [γi (G), G] for i = 1, 2, . . . implies that (9.4) Li+1 = [Li , L] for all integers i ≥ 1. Hence the additive subgroup L1 generates the Lie ring L. Finally, the commutator equation [x, x] = 1 for all x ∈ G implies that (9.5) [xi , xi ] = 0 in L for any i ≥ 1 and any xi ∈ Li . This and the standard Lie identity [x, y] = −[y, x] for all x, y ∈ L easily imply that [x, x] = 0 for all x ∈ L. Since each component Li of L is a finite additive p-group, its Frattini subgroup Φ(Li ) is just pLi . The factor group (9.6) L̄i := Li /pLi = Li /Φ(Li ) is now a vector space over the finite field Fp of p elements. After natural identifications, this vector space is the i-th component in the graded Lie algebra (9.7) L̄ := L/pL = (L1 /pL1 ) ⊕ (L2 /pL2 ) ⊕ . . . = L̄1 ⊕ L̄2 . . . over the field Fp . The equation (9.4) implies that (9.8) L̄i+1 = [L̄i , L̄1 ] for all i ≥ 1. Hence the subspace L̄1 generates the Lie algebra L̄ over Fp . Finally, the equation (9.5) implies that (9.9) [x̄i , x̄i ] = 0 in L̄ for all xi ∈ L̄i , where i ≥ 1. For a vector space V over the finite field Fp , dim(V ) denotes the dimension of V . For the rest of this section, for the Lie algebra L̄, we assume that (9.10) dim(L̄3 ) (9.11) [L̄1 , ȳ] = 1 = L̄2 and for all ȳ ∈ L̄1 − {0}. CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 21 Following the notation of Macdonald in [14], we denote by m and n the dimensions (9.12) dim(L̄1 ) = m (9.13) dim(L̄2 ) = n and of L̄1 and L̄2 respectively, over Fp . It follows from (9.8) and (9.10) that [L̄2 , L̄1 ] = L̄3 is a nonzero vector space over Fp . Hence neither L̄1 nor L̄2 can be zero and therefore their dimensions m and n satisfy (9.14) m, n > 0. An m × m square matrix A = (ai,j ) over Fp is called strongly skew-symmetric if ai,j = −aj,i whenever i 6= j and ai,i = 0 for all i. Notice that when p is odd, a matrix is strongly skewsymmetric if and only if it is skew-symmetric. But when p = 2, a matrix is skew-symmetric if and only if it is symmetric, which is not true for strongly skew-symmetric matrices. Thus the notion of strongly skew-symmetric matrix is different from a skew-symmetric one when p = 2. Important thing here is that Pfaffians exist only for strongly skew-symmetric matrices, which is a crucial ingredient in the following proof. Now we are going to reproduce Macdonald’s arguments [14, pages 353-355] in additive setup to prove the following. Lemma 9.1. If m and n are as above, then m is even and m ≥ 2n. Proof. Let L̄1 × L̄1 → L̄2 be the Lie product function. Let x̄1 , x̄2 , . . . , x̄m and ȳ1 , ȳ2 , . . . , ȳn be the bases of L̄1 and L̄2 respectively. Then (9.15) [x̄i , x̄j ] = αi,j,1 ȳ1 + · · · + αi,j,m ȳn for 1 ≤ i, j ≤ m, where αi,j,k ∈ Fp for 1 ≤ k ≤ n. Let Ak be the matrix (αj,i,k ). It follows from the defintion of Lie product and (9.9) that Ak is a strongly skew-symmetric matrix. Let x̄ = φ1 x̄1 + φ2 x̄2 + · · · + φm x̄m be an arbitrary non-trivial element of L̄1 , where φi ∈ Fp for 1 ≤ i ≤ m. Notice that all of φi can not be zero as x̄ is non-trivial element of L̄1 . Since Lie product is Fp -bilinear, we have [x̄, x̄i ] = φ1 [x̄1 , x̄i ] + · · · + φm [x̄m , x̄i ] = φ1 (α1,i,1 ȳ1 + · · · + α1,i,m ȳn ) + · · · + φm (αm,i,1 ȳ1 + · · · + αm,i,m ȳn ) = βi,1 ȳ1 + · · · + βi,n ȳn where βi,j = α1,i,j φ1 + · · · + αm,i,j φm . Let B denote the matrix (βi,j ) for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Notice that B is an m × n matrix and B = (A1 Φ, . . . , An Φ), where Φ is a column matrix whose transpose ΦT is the row matrix (φ1 , φ2 , . . . , φm ). For any non-trivial element x̄ ∈ L̄1 , we have a Fp -linear map [·, x̄] : L̄1 × L̄1 → L̄2 which maps any ū ∈ L̄1 to [ū, x̄]. Now (9.11) tells that this map is surjective. Hence m = dim(L̄1 ) > dim(L̄2 ) = n, because the kernel of [·, x̄] is non-trivial. Also, since [·, x̄] is surjective and {x̄1 , x̄2 , . . . , x̄m } is a basis for L̄1 , it follows that the set of images [x̄i , x̄] of the elements x̄i for 1 ≤ i ≤ m contains a linearly independent subset of cardinality n. Thus it follows that CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 22 rank(B), the rank of the matrix B, is equal to n (this happens because the image [x̄, x̄i ] constitutes ith row of the matrix B). Hence the matrix B has n linearly independent columns. But there are only n columns in B. Therefore all the columns A1 Φ, A2 Φ, . . . , An Φ are linearly independent. As mentioned above, each Ai is skew-symmetric. Therefore A := ψ1 Ai + ψ1 A2 + · · · + ψn An is also skew-symmetric for any choice of ψ1 , ψ2 , . . . , ψn in Fp . Suppose that not all of the ψ’s are zero. If the determinant of A is zero, then the determinant of AΦ = ψ1 A1 Φ + ψ2 A2 Φ + · · · + ψn An Φ is also zero. This implies that the determinant of B is zero. But this is a contradiction to the fact that A1 Φ, A2 Φ, . . . , An Φ are linearly independent. Hence it follows that if not all of the ψ’s are zero, then A is non-singular. As the degree of the square matrix A is m and all skew-symmetric matrices of odd degree are singular, it follows that m is even. Set m = 2d. Now consider the equation |A| = 0, where |A| denotes the determinant of the matrix A. The left hand side of the equation is a polynomial f := f (ψ1 , ψ2 , . . . , ψn ) of total degree m = 2d in n unknowns ψ1 , ψ2 , . . . , ψn in Fp . Since A is skew-symmetric, there exists a Pfaffian g := g(ψ1 , ψ2 , . . . , ψn ) of A. Thus g 2 = f , and f is homogeneous. So we get a homogeneous polynomial equation f = 0 of total degree d in n variables. We claim that n ≤ d. For, if n > d, then it follows from Chevalley-Warning theorem [9] that the number of solution of f = 0 in Fp is divisible by p. Since g is homogeneous, it has 0 as a solution, and therefore g has a non-zero solution. This contradicts the fact that A is non-singular. Hence our claim is true. This implies that m = 2d ≥ 2n, which completes the proof of the lemma. . Define C̄ to be the Fp -subspace (9.16) C̄ := {ȳ ∈ L̄1 | [L̄2 , ȳ] = 0} in L̄1 . Lemma 9.2. If x̄ ∈ L̄1 − C̄ and ȳ ∈ C̄ − {0}, then [x̄, ȳ] 6= 0 in the Lie algebra L̄. Proof. Let z̄ be an element of L̄1 . Since the nilpotency class of our Lie algebra L̄ is three, the Jacobi identity gives [[x̄, ȳ], z̄][[z̄, x̄], ȳ][[ȳ, z̄], x̄] = 0. Here [z̄, x̄] ∈ [L̄1 , L̄1 ] = L̄2 and ȳ ∈ C̄. Thus by (9.16) [[z̄, x̄], ȳ] = 0. Hence the above equation is equavalent to [[x̄, ȳ], z̄] = −[[ȳ, z̄], x̄]. Because x̄ ∈ L̄1 does not lie in C̄, by the definition of C̄ there exists an element w̄ ∈ L̄2 such that [w̄, x̄] 6= 0. Our hypothesis (9.11) implies that [ȳ, L̄1 ] = −[L̄1 , ȳ] = L̄2 for the non-zero element ȳ of L̄1 . Hence we can choose z̄ ∈ L̄1 so that [ȳ, z̄] = w̄. Then the above equation becomes [[x̄, ȳ], z̄] = −[w̄, x̄] 6= 0. Hence [x̄, ȳ] 6= 0 and the proof is complete.  CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 23 Notice that the Fp -linear maps from L̄2 to L̄3 naturally form a vector space Hom(L̄2 , L̄3 ) over Fp . It follows from (9.10) and (9.13) that this vector space has dimension (9.17) dim(Hom(L̄2 , L̄3 )) = dim(L̄2 ) · dim(L̄3 ) = n · 1 = n. Lie multiplication in L̄ is Fp -bilinear from L̄2 × L̄1 to L̄3 . Hence it induces a Fp -linear map λ : L̄1 → Hom(L̄2 , L̄3 ), sending any ȳ in L̄1 to a linear map λ(ȳ) ∈ Hom(L̄2 , L̄3 ) defined by (9.18) λ(ȳ) : x̄ 7→ [x̄, ȳ] ∈ L̄3 for any x̄ ∈ L̄2 . By the definition (9.16) of C̄ it follows that C̄ is precisely the kernel (9.19) ker(λ) = C̄ of λ. Hence we have (9.20) m − dim(C̄) = dim(L̄1 /C̄) ≤ dim(Hom(L̄2 , L̄3 )) = n, where equality holds if and only if λ sends L̄1 onto Hom(L̄2 , L̄3 ). Lemma 9.3. The subspace C̄ is a non-zero proper subspace of the vector space L̄1 . Proof. By the definition (9.16) of C̄, it is a subspace of L̄1 . If C̄ = 0, then dim(C̄) = 0. Thus by Lemma 9.1 and (9.20) we get 2n ≤ m ≤ n, which is impossible because n > 0 by (9.14). Hence C̄ is a non-zero subspace of L̄1 . If C̄ = L̄1 , then (9.8) and (9.16) imply that L̄3 = [L̄2 , L̄1 ] = [L̄2 , C̄] = 0. But this contradicts (9.10). Hence C̄ is a proper subspace of L̄1 , and the proof is complete.  Notice that any element z̄ ∈ L̄1 determines an Fp -linear map [·, z̄] : x̄ 7→ [x̄, z̄] of L̄1 into L̄2 . The kernel of this map is the centralizer (9.21) C(z̄) = CL̄1 (z̄) := {x̄ ∈ L̄1 | [x̄, z̄] = 0} of z̄ in L̄1 . If z̄ 6= 0, then (9.11) implies that [·, z̄] is an epimorphism of the vector space L̄1 onto L̄2 . Thus it follows that (9.22) dim(C(z̄)) = dim(L̄1 ) − dim(L̄2 ) = m − n for all z̄ in L̄1 − {0}. Lemma 9.4. If z̄ ∈ C̄ − {0}, then C(z̄) ⊆ C̄. If z̄ ∈ L̄1 − C̄, then C(z̄) ∩ C̄ = {0}. Proof. Suppose that z̄ in C̄ − {0}, and x̄ ∈ C(z̄) − C̄. Then [x̄, z̄] = 0 by (9.21). But Lemma 9.2 for ȳ = z̄ implies that [x̄, z̄] 6= 0. This contradiction proves the first statement in the lemma. Now assume that z̄ in L̄1 − C̄, and that ȳ ∈ C(z̄)∩ C̄ is not 0. Then by (9.21) [z̄, ȳ] = −[ȳ, z̄] = 0. But again Lemma 9.2 for x̄ = z̄ implies that [z̄, ȳ] 6= 0. Thus we again get a contradiction, and therefore C(z̄) ∩ C̄ = {0}. This completes the proof.  CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 24 Proposition 9.5. If the Lie algebra L̄ satisfies (9.10) and (9.11), then (a) (b) m = 2n. If z̄ ∈ L̄1 − {0}, then the subspace C(z̄) of L̄1 has dimesion n. (c) The subspace C̄ of L̄1 has dimension n, and is equal to C(ȳ) for any ȳ ∈ C̄ − {0}. Hence λ is an epimorphism of L̄1 onto Hom(L̄2 , L̄3 ) with C̄ as its kernel. (d) If x̄ ∈ L̄1 − C̄, then L̄1 is the direct sum C(x̄) ⊕ C̄ of its subspaces C(x̄) and C̄. Hence the map z̄ 7→ [x̄, z̄] is an isomorphism of C̄ onto L̄2 as Fp -spaces. Proof. In view of Lemma 9.3 we can choose some elements x̄ ∈ L̄1 − C̄ and ȳ ∈ C̄ − {0}. Then it follows from Lemma 9.4 that the subspaces C(x̄) and C(ȳ) satisfy C(x̄)∩ C̄ = {0} and C(ȳ) ⊆ C̄. Hence C(x̄) ∩ C(ȳ) = {0}. It follows that the direct sum C(x̄) ⊕ C(ȳ) of these two subspaces is a subspace of L̄1 . In view of (9.12) and (9.22), this implies that 2(m − n) = dim(C(x̄) ⊕ C(ȳ)) ≤ dim(L̄1 ) = m, where equality holds if and only if L̄1 = C(x̄) ⊕ C(ȳ). But m − n ≥ n by Lemma 9.1. Therefore 2(m − n) ≥ n + (m − n) = m. So the above inequality is an equality, and L̄1 is the direct sum (9.23) L̄1 = C(x̄) ⊕ C(ȳ) of the subspaces C(x̄) and C(ȳ). Part (a) of the proposition is nothing but 2(m − n) = m. Part (a) and (9.22) imply part (b). Because C(x̄) ∩ C̄ = {0}, the sum C(x̄) + C̄ of subspaces of L̄1 is direct. But L̄1 is already the direct sum (9.23), in which the summand C(ȳ) is contained in C̄. This is possible only when C(ȳ) = C̄. Hence dim(C̄) = dim(C(ȳ)) = n, and C̄ = C(ȳ) for all ȳ ∈ C̄ − {0}. It follows that m − dim(C̄) = m − n = n. Thus equality holds in (9.20). This implies that λ sends L̄1 onto Hom(L̄2 , L̄3 ). Since ker(λ) = C̄ by (9.19), this completes the proof of part (c) in the proposition. The decomposition (9.23) and the equality C̄ = C(ȳ) imply that L̄1 is the direct sum C(x̄)⊕ C̄ for any x̄ ∈ L̄1 − C̄. It follows from (9.11) and (9.21) that C(x̄) is the kernel of the epimorphism z̄ 7→ [x̄, z̄] of L̄1 onto L̄2 . Hence this epimorphism restricts to an isomorphism of C̄ onto L̄2 . The part (d) holds, and the proposition is proved.  10. Proof of Theorems C and D Let us recall that for an arbitrary finite p-group G, for a prime p, L denotes the graded Lie ring defined in (9.1), and L̄ denotes its factor Lie algebra defined in (9.7). For any integer i ≥ 1, the additive group Li is just the multiplicative group γi (G)/γi+1 (G) written additively. Hence Φ(Li ) = pLi is the subgroup Φ(γi (G)/γi+1 (G)) = Φ(γi (G))γi+1 (G)/γi+1 (G) ≤ γi (G)/γi+1 (G) written additively. It follows that there is a natural epimorphism ηi of the multiplicative group γi (G) on to the additive group L̄i = Li /Φ(Li ), sending any x ∈ γi (G) to (10.1) ηi (x) := (xγi+1 (G))Φ(Li ) ∈ Li /Φ(Li ) = L̄i . Furthermore, this epimorphism has the kernel (10.2) ker(ηi ) = Φ(γi (G))γi+1 (G). CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 25 Since Lie multiplication in L̄ = L/pL is induced by that in L, the definition (9.3) of the latter multiplication implies that (10.3) [ηi (x), ηj (y)] = ηi+j ([x, y]) in L̄ for any integers i, j ≥ 1, and any elements x ∈ γi (G) and y ∈ γj (G). Here the commutator [x, y] is computed in the group G, and lies in [γi (G), γj (G)] ≤ γi+j (G). So the right hand side of this equation is well defined. The situation when i = 1 is a bit special, since γi (G) is G itself, and γ2 (G) is a subgroup of Φ(G) = Φ(γ1 (G)). It follows that the kenel Φ(γ1 (G))γ2 (G) of η1 is just Φ(G). Hence we have Proposition 10.1. The epimorphism η1 in (10.1) sends the multiplicative group G onto the additive group L̄1 , and has Φ(G) as its kernel. So elements x1 , x2 , . . . , xd ∈ G form a minimal generating set for the p-group G if and only if their images η1 (x1 ), η1 (x2 ), . . . , η1 (xd ) form a basis of the vector space L̄1 over Fp . Hence the number d = d(G) of elements in any minimal generating set for G is the dimension dim(L̄1 ) of the Fp -space L̄1 . Since L1 is G/γ2 (G) written additively, and since L1 generates the Lie ring L, an automorphism α ∈ Aut(G) lies in Autγ2 (G) (G), the group of automorphisms of G which induce the identity on G/γ2 (G), if and only if it induces the identity automorphism on L. In particular, any such α induces the identity automorphism on the section γi (G)/γi+1 (G), which is Li written multiplicatively, for any integer i ≥ 1. Suppose that {x1 , x2 , . . . , xd } is a minimal generating set for G, and that α ∈ Aut(G). Then there are unique elements y1 , y2 , . . . , yd ∈ G such that α(xi ) = xi yi for 1 ≤ i ≤ d. Evidently these elements yi determine α completely. Furthermore, α lies in the subgroup Autγ2 (G) (G) of Aut(G) if and only if yi ∈ γ2 (G) for 1 ≤ i ≤ d. It follows that | Autγ2 (G) (G)| ≤ |γ2 (G)|d , with equality if and only if there is some automorphism of the groups G sending xi to xi yi for any choice of the elements yi in γ2 (G) for 1 ≤ i ≤ d. Hence we have Proposition 10.2. If p is any prime, then the following properties are equivalent for any finite p-group G: (10.4a) | Autγ2 (G) (G)| = |γ2 (G)|d . (10.4b) If {x1 , x2 , . . . , xd } is any minimal generating set for G, and if y1 , y2 , . . . , yd be any d elements in γ2 (G), then there exists an automorphism α of G sending xi to xi yi for all 1 ≤ i ≤ d. (10.4c) There is some minimal generating set {x1 , x2 , . . . , xd } for G such that, for any y1 , y2 , . . . , yd ∈ γ2 (G), there is some automorphism α of G sending xi to xi yi for all 1 ≤ i ≤ d. For the rest of this section we assume that the finite p-group satisfies (10.5a) |γ3 (G)| = p, (10.5b) (10.5c) [G, y] = γ2 (G) | Aut γ2 (G) for all y ∈ G − Φ(G), and (G)| = |γ2 (G)|d(G) . As a consequence G also satisfies CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 26 Lemma 10.3. The kernel Φ(γ3 (G))γ4 (G) of the epimorphism η3 : γ3 (G) ։ L̄3 is 1. Hence G has class 3, and η3 is an isomorphism of the multiplicative group γ3 (G) onto the additive group L̄3 . So dim(L̄3 ) = 1. Proof. Since the subgroup γ3 (G) is of order p (by (10.5a)), it follows that γ3 (G) ≤ Z(G) and Φ(γ3 (G)) = 1. Hence G is of class 3 and by (10.2) η3 is an isomorphism.  The last statement in the preceding lemma tells us that L̄ satisfies (9.10). Since η2 is an epimorphism of γ2 (G) onto L̄2 , it follows from (10.3) and (10.5b) that (9.11) holds. Hence all the hypotheses of Proposition 9.5 hold. So from now on we adopt the notation m, n, C̄, C(z̄), λ used in Proposition 9.5, and apply freely its conclusions (a) through (d). Following Macdonald, we denote by C the centralizer (10.6) C = CG (γ2 (G)) of γ2 (G) in G. It has another description. Lemma 10.4. The subgroup C of G is the inverse image of the subspace C̄ of L̄1 under the epimorphism η1 of G = γ1 (G) onto L̄1 . Hence C contains the kernel Φ(G) of the epimorphism. Proof. If x ∈ γ2 (G) and y ∈ G, then (10.3) tells us that η3 sends [x, y] ∈ γ3 (G) to [η2 (x), η1 (y)] ∈ L̄3 . Since η3 is an isomorphism of γ3 (G) onto L̄3 , it follows that [x, y] = 1 in G if and only if [η2 (x), η1 (y)] = 0 in L̄3 . Since η2 is an epimorphism of γ2 (G) onto L̄2 , we conclude that y ∈ G satisfies [x, y] = 1 for all x ∈ γ2 (G) if and only if its image η1 (y) ∈ L̄1 satisfies [x̄, η1 (y)] = 0 in L̄ for all x̄ ∈ L̄2 . In view of (9.16) and (10.6) this proves the first statement of the lemma. Since ker(η1 ) = Φ(G) by Proposition 10.1, the remaining part of the statement also holds true.  Now we are ready to prove the key result. Theorem 10.5. If a finite p-group, for any prime p, satisfies (10.5a)-(10.5c), then d(G) = 2. Proof. The number d(G) is equal to the dim(L̄1 ) by Proposition 10.1. This, (9.12), (9.13) and Proposition 9.5(a) imply that d(G) = m = 2n. So we only need to show that n = 1. Since the integer n > 0 by Lemma 9.1, it suffices to derive a conradiction from the assumption that n ≥ 2. The subspace C̄ of L̄1 has dimension n by Proposition 9.5(c). Since L̄1 has dimension m = 2n, we can choose elements x̄1 , x̄2 , . . . , x̄n ∈ L̄1 −C̄ forming a basis of L̄1 modulo C̄, i.e., whose images form a basis for the factor group L̄1 /C̄. Proposition 9.5(c) implies that λ(x̄1 ), λ(x̄2 ), . . . , λ(x̄n ) form a Fp -basis for Hom(L̄2 , L̄3 ). Let w̄ be a basis element of the vector space L̄3 of dimension 1. Then there is a unique basis z̄1 , z̄2 , . . . , z̄n for L̄2 such that λ(x̄i ) sends z̄j to δij w̄ for 1 ≤ i, j ≤ n, where δij is the Kronecker delta symbol equal to 1 when i = j and 0 otherwise. In view of the definition (9.18) of λ, this says that (10.7) [z̄j , x̄i ] = δij w̄ ∈ L̄3 for any 1 ≤ i, j ≤ n. Since x̄1 lies in L̄1 − C̄, by Proposition 9.5(d) it follows that the map ȳ 7→ [x̄1 , ȳ] is an isomorphism of the vector space C̄ onto L̄2 . Hence there is a unique basis ȳ1 , ȳ2 , . . . , ȳn for C̄ such that (10.8) [x̄1 , ȳj ] = z̄j CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 27 for all 1 ≤ j ≤ n. By definition x̄1 , x̄2 , . . . , x̄n form a basis for L̄1 /C̄, while ȳ1 , ȳ2 , . . . , ȳn form a baisis for the subspace C̄. Hence x̄1 , x̄2 , . . . , x̄n , ȳ1 , ȳ2 , . . . , ȳn form a basis for L̄1 . In view of Proposition 10.1 there is some minimal family of generators x1 , x2 , . . . , xn , y1 , y2 , . . . yn for the p-group G such that η1 (xi ) = x̄i and η1 (yj ) = ȳj for all 1 ≤ i, j ≤ n. Since [x1 , y1 ] lies in γ2 (G), Proposition 10.2 and our assumption (10.5c) imply that there is some automorphism α ∈ Autγ2 (G) (G) sending these generators to (10.9) α(xi ) = xi , α(y1 ) = y1 [x1 , y1 ], α(yj ) = yj for 1 ≤ i ≤ n and 2 ≤ j ≤ n. We claim that the automorphism α fixes every element of γ2 (G). Our assumption that n ≥ 2 implies that the element y2 exists and is fixed by α. Since α also fixes x1 , x2 , . . . , xn , it fixes each commutator [xi , y2 ] ∈ γ2 (G) for 1 ≤ i ≤ n. We know from (10.3) that η2 ([xi , y2 ]) = [η1 (xi ), η1 (y2 )] = [x̄i , ȳ2 ] for all 1 ≤ i ≤ n. Because ȳ2 lies in C̄ − {0}, its centralizer C(ȳ2 ) is C̄ by Proposition 9.5(c). So the basis x̄1 , x̄2 , . . . , x̄n for L̄1 modulo C̄ is also a basis for L̄1 modulo C(ȳ2 ). The epimorphism x̄ 7→ [x̄, ȳ2 ] of L̄1 onto L̄2 has kernel C(ȳ2 ). Hence it sends the x̄i , for 1 ≤ i ≤ n, to a basis of L̄2 consisting of all [x̄i , ȳ2 ] = η2 ([xi , y2 ]). We conclude that [x1 , y2 ], [x2 , y2 ], . . . , [xn , y2 ] generate γ2 (G) modulo γ3 (G). But α, like any automorphism in Autγ2 (G) (G), fixes each element of γ3 (G) = γ3 (G)/γ4 (G). Since it also fixes each [xi , y2 ] for 1 ≤ i ≤ n, it fixes every elements of γ2 (G), and our claim is proved. Now we consider the action of α on the commutator [x1 , y1 ]. Since the commutator lies in γ2 (G), it must be fixd by α. In view of (10.9) this gives [x1 , y1 ] = = α([x1 , y1 ]) = [α(x1 ), α(y1 )] = [x1 , y1 [x1 , y1 ]] [x1 , [x1 , y1 ]][x1 , y1 ][x1 ,y1 ] = [x1 , [x1 , y1 ]][x1 , y1 ] in G. Hence the element [x1 , [x1 , y1 ]] = 1 in γ3 (G). Applying η3 , and using (10.3) twice, we obtain [x̄1 , [x̄1 , ȳ1 ]] = [η1 (x1 ), [η1 (x1 ), η1 (y1 )]] = η3 ([x1 , [x1 , y1 ]]) = 0 in L̄. But [x̄1 , [x̄1 , ȳ1 ]] = [x̄1 , z̄1 ] = −w̄ 6= 0 by (10.8) and (10.7). This contradiction shows that our assumption n ≥ 2 must be false. Hence n = 1, which completes the proof of the Theorem.  Now we are ready to prove Theorems C and D. Proof of Theorem C. Let G be a Camina-type finite p-group of nilpotency class at least 3. Then there exists a maximal subgroup N of γ3 (G) which is normal in G. Set Ḡ = G/N . Notice that Ḡ is Camina-type (by Lemma 4.4), the nilpotency class of Ḡ is 3 and |γ3 (Ḡ)| = p. Now it follows from Proposition 10.1 and the preceding discussion for Ḡ that d(Ḡ) = dim(L̄1 ) and d(γ2 (Ḡ)/γ3 (Ḡ)) = dim(L̄2 ). This, (9.12), (9.13) and Proposition 9.5(a) imply that d(Ḡ) = 2d(γ2 (Ḡ)/γ3 (Ḡ). Since d(Ḡ) = d(G) and d(γ2 (Ḡ)/γ3 (Ḡ)) = d(γ2 (G)/γ3 (G)), we have d(G) = CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 28 2d(γ2 (G)/γ3 (G)). This proves assertion (i). Assertions (ii) and (iii) follow from Theorem 6.4 and Corollary 7.3 respectively.  Proof of Theorem D. Let G be a finite p-group of nilpotency class at least 3 satisfying Hypothesis A. Let N be a maximal subgroup of γ3 (G) which is normal in G. Set Ḡ = G/N . Then Ḡ satisfies Hypothesis A (by Lemma 4.3), the nilpotency class of Ḡ is 3 and |γ3 (Ḡ)| = p. Notice that Hypothesis A is stronger than (10.5c), so that any group satisfying Hypothesis A, also satisfies (10.5c). So all the hypotheses of Theorem 10.5 hold true for Ḡ. Hence d(Ḡ) = 2, and hence d(G) = 2. This completes the proof of assertion (i) of the theorem. Assertion (ii) follows from assertion (i), Lemma 7.1, Lemma 7.2, Proposition 7.4 and Theorem 7.5. Assertion (iii) follows from Theorem 8.3.  11. Central quotient and commutator subgroup Understanding the relationship between γ2 (G) and G/ Z(G) goes back, at least, to 1904 when I. Schur [18] proved that the finiteness of G/ Z(G) implies the finiteness of γ2 (G). A natural question which arises here is about the converse of Schur’s theorem, i.e., whether the finiteness of γ2 (G) implies the finiteness of G/ Z(G). By a well known result of P. Hall [12] it follows that if γ2 (G) is finite, then G/ Z2 (G) is finite. However, unfortunately, the answer to the converse of Schur’s theorem in general is negative as it can be seen for infinite extraspecial p-group for an odd prime p, results are available with some extra conditions. We mention here one of these results, which is generalised below. B. H. Neumann [17, Corollary 5.41] proved that if G is finitely generated and γ2 (G) is finite, then G/ Z(G) is finite. Moreover, if G is generated by k elements, then |G/ Z(G)| ≤ |γ2 (G)|k . We notice that neither of the conditions, i.e., G is finitely generated and γ2 (G) finite, is necessary to show that G/ Z(G) is finite. Let G/ Z(G) be finitely generated by a minimal generating set {x1 Z(G), . . . , xd Z(G)} having d = d(G) elements such that |xG i | is finite for 1 ≤ i ≤ d. Since any class-preserving automorphism of G fixes the center of G elementwise and maps the non-central generating elements to their conjugates, it follows that | Autc (G)| ≤ Πdi=1 |xG i |. d G Thus |G/ Z(G)| = | Inn(G)| ≤ | Autc (G)| ≤ Πi=1 |xi | is finite, which implies that γ2 (G) is finite. Hence it follows that (11.1) |G/ Z(G)| ≤ |γ2 (G)|d , since |g G | = |[g, G]| ≤ |γ2 (G)| for all g ∈ G. We have proved Proposition 11.1. Let G be an arbitrary group such that G/ Z(G) is finitely generated by a minimal generating set {x1 Z(G), . . . , xd Z(G)} with |xG i | < ∞ for 1 ≤ i ≤ d. Then G/ Z(G) is finite. Moreover, γ2 (G) is finite and |G/ Z(G)| ≤ |γ2 (G)|d . A natural question which arises here is: Question. What are all the groups G for which equality holds in (11.1)? We are going to consider only nilpotent groups G for which equality holds in (11.1). In this case, there exists a finite nilpotent group H such that Z(H) ≤ γ2 (H). Then it follows that d(H) = d(H/ Z(H)) = d(G/ Z(G). So we work with finite nilpotent groups G such that CLASS-PRESERVING AUTOMORPHISMS OF FINITE p-GROUPS II 29 Z(G) ≤ γ2 (G), and provide an answer to the above question upto isoclinism. Let G be a finite nilpotent group minimally generated by d elements for which equality holds in (11.1). Then it follows that |G/ Z(G)| = | Inn(G)| ≤ | Autc (G)| ≤ |γ2 (G)|d = G/ Z(G). Hence | Autc (G)| = |γ2 (G)|d , which shows that G satisfies Hypothesis A. Notice that Autc (G) = Inn(G). Since G satisfies Hypothesis A, it is a Camina-type group. Thus it follows from Lemma 3.4 that G is a p-group for some prime integer p. So the problem now reduces to classifying finite p-groups G for which equality holds in (11.1). Obvious examples of such groups are finite extraspecial p-groups. Other examples are groups defined in (1.3) and 2-generator groups isoclinic to these groups. For finite p-groups of nilpotency class 2, we get Theorem 11.2. Let G be a finite p-group of nilpotency class 2 such that equality holds in (11.1). Then G is isoclinic to the group Y defined in (5.2). Proof. Suppose that G is a group as in the statement. Since Autc (G) = Inn(G) (as noticed above), it follows from Theorem 5.10 that γ2 (G) is cyclic. Since G satisfies Hypothesis A, by Theorem 5.11 G is isoclinic to the group Y defined in (5.2).  For finite p-groups of nilpotency class larger than 2, we get Theorem 11.3. Let G be a finite p-group of nilpotency class at least 3 for which equality holds in (11.1). Then d(G) = 2. If |γ2 (G)/γ3 (G)| > 2, then equality holds for G in (11.1) if and only if G is a 2-generator group with cyclic commutator subgroup. Furthermore, G is isomorphic to some group defined in (7.2) and is isoclinic to the group K defined in (1.3) for suitable parameters. If |γ2 (G)/γ3 (G)| = 2, then equality holds for G in (11.1) if and only if G is a 2-generator 2-group of nilpotency class 3 with elementary abelian γ2 (G) of order 4. Proof. Since any group G, as in the statement, satisfies Hypothesis A, the proof of all assertions, except the last one, follows from Theorem D stated in the introduction. The last assertion follows from Proposition 8.4.  Acknowledgements. I thank Prof. Everett Dade for his timely help and valuable suggestions. Results of Sections 9 and 10 are suggested by him. 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