In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐ design, both the bloc... more In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐ design, both the block‐size and the number of points are bounded by functions of , but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of bounding and . For we obtain a list of “numerically feasible” parameter sets together with the number of parts and part‐size of an invariant point‐partition and the size of a nontrivial block‐part intersection. Moreover from these parameter sets we determine all examples with fewer than 100 points. There are exactly 11 such examples, and for one of these designs, a flag‐regular, point‐imprimitive design with automorphism group , there seems to be no construction previously available in the literature.
Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-impri... more Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer–Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer–Doyen parameters.
The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum di... more The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F2 for finite 2-transitive permutation groups.
In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the ... more In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,\delta)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,\delta)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) Nordstrom-Robinson code.
Page 303. Cyclic matrices and the meataxe Peter M. Neumann and Cheryl E. Praeger Abstract. We pre... more Page 303. Cyclic matrices and the meataxe Peter M. Neumann and Cheryl E. Praeger Abstract. We present variants of two of the procedures for analysing matrix represen-tations of groups which form part of the so-called MEATAXE collection of algorithms. ...
Journal of the Australian Mathematical Society, 2016
This article began as a study of the structure of infinite permutation groups $G$ in which point ... more This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1985
Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. Thi... more Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.
Bulletin of the Australian Mathematical Society, 1978
Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the... more Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the stabilizer Gα of αhas a set σ = {B1, …, Bt} of nontrivial blocks of imprimitivity in Ω – {α}. If Gα is 3-transitive on σ it is shown that either G is a collineation group of a desarguesian projective or affine plane or no nonidentity element of Gα fixes B pointwise.
In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐ design, both the bloc... more In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐ design, both the block‐size and the number of points are bounded by functions of , but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of bounding and . For we obtain a list of “numerically feasible” parameter sets together with the number of parts and part‐size of an invariant point‐partition and the size of a nontrivial block‐part intersection. Moreover from these parameter sets we determine all examples with fewer than 100 points. There are exactly 11 such examples, and for one of these designs, a flag‐regular, point‐imprimitive design with automorphism group , there seems to be no construction previously available in the literature.
Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-impri... more Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer–Doyen parameters, linking the block size with the parameters of an associated imprimitivity system on points. We show that the Delandtsheer–Doyen parameters provide upper bounds on the permutation ranks of the groups induced on the imprimitivity system and on a class of the system. We explore extreme cases where these bounds are attained, give a new construction for a family of designs achieving these bounds, and pose several open questions concerning the Delandtsheer–Doyen parameters.
The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum di... more The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F2 for finite 2-transitive permutation groups.
In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the ... more In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,\delta)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,\delta)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) Nordstrom-Robinson code.
Page 303. Cyclic matrices and the meataxe Peter M. Neumann and Cheryl E. Praeger Abstract. We pre... more Page 303. Cyclic matrices and the meataxe Peter M. Neumann and Cheryl E. Praeger Abstract. We present variants of two of the procedures for analysing matrix represen-tations of groups which form part of the so-called MEATAXE collection of algorithms. ...
Journal of the Australian Mathematical Society, 2016
This article began as a study of the structure of infinite permutation groups $G$ in which point ... more This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.
Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1985
Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. Thi... more Generalised wreath products of permutation groups were discussed in a paper by Bailey and us. This note determines the orbits of the action of a generalised wreath product group on m–tuples (m ≥ 2) of elements of the product of the base sets on the assumption that the action on each component is m–transitive. Certain related results are also provided.
Bulletin of the Australian Mathematical Society, 1978
Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the... more Suppose that G is a doubly transitive permutation group on a finite set Ω and that for α in ω the stabilizer Gα of αhas a set σ = {B1, …, Bt} of nontrivial blocks of imprimitivity in Ω – {α}. If Gα is 3-transitive on σ it is shown that either G is a collineation group of a desarguesian projective or affine plane or no nonidentity element of Gα fixes B pointwise.
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Papers by Cheryl Praeger