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    Math 55a Pset1

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    benjy321
    This document provides the homework assignment for Math 55a in the Fall 2018 semester. The homework is due on September 12th and contains 11 problems covering topics in set theory, group theory, and homomorphisms. It also provides two supplementary problems from Artin's textbook.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Math 55a Pset1

    Uploaded by

    benjy321
    21 views1 page
    This document provides the homework assignment for Math 55a in the Fall 2018 semester. The homework is due on September 12th and contains 11 problems covering topics in set theory, group theory, and homomorphisms. It also provides two supplementary problems from Artin's textbook.

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    pset1 55a

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    This document provides the homework assignment for Math 55a in the Fall 2018 semester. The homework is due on September 12th and contains 11 problems covering topics in set theory, group theory, and homomorphisms. It also provides two supplementary problems from Artin's textbook.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    21 views1 page

    Math 55a Pset1

    Uploaded by

    benjy321
    This document provides the homework assignment for Math 55a in the Fall 2018 semester. The homework is due on September 12th and contains 11 problems covering topics in set theory, group theory, and homomorphisms. It also provides two supplementary problems from Artin's textbook.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    Homework 1

    Math 55a, Fall 2018

    Due Wednesday, September 12, 2018

    Note: the first three problems are warm-up problems on set theory; if you don’t know how to
    do them, ask us!
    1. Prove the pigeon-hole principle: if A is a finite set, then any injective map f : A → A is also
    surjective.
    2. Let N = {0, 1, 2, . . .} denote the set of natural numbers. Give an explicit bijection between N
    and N × N.
    3. Let F denote the set of all functions f : R → R, and let C ⊂ F denote the subset of all
    continuous functions. Prove that |R| = |C| < |F |.
    4. Let x1 , x2 , . . . , xn ∈ G be any elements of a group G. Show that
    −1 −1 −1
    (x1 x2 · · · xn )−1 = x−1
    n · xn−1 · · · x2 · x1 .

    5. Show that a group G cannot be the union of two proper subgroups.


    6. Show that any finite group G of even order contains an element x ∈ G such that a 6= e but
    a2 = e.
    7. Let D8 be the group of symmetries of a square (including reflections). How many subgroups
    (including the trivial subgroups D8 and {e}) does D8 have?
    8. Let G be a group, and consider the set map φ : G → G sending each element a ∈ G to its
    square a2 ∈ G. Show that φ is a homomorphism if and only if G is abelian.
    9. Let H ⊂ G be any subgroup of a finite group G. Show that if |G|/|H| = 2 then H is a normal
    subgroup of G.
    10. What is the order of the group GL2 (Z/3) of 2 × 2 matrices with entries in Z/3 and nonzero
    determinant?
    11. Let G be a group.
    (a) Show that the set of automorphisms of G is itself a group (with group law given by
    composition). This group is denoted Aut(G).
    (b) For each element a ∈ G, define a map ca : G → G by ca (x) = axa−1 . Show that ca is an
    automorphism of G.
    (c) Show that the map φ : G → Aut(G) defined by sending a ∈ G to ca ∈ Aut(G) is a
    homomorphism.
    (d) Give an example of a group G such that φ is an isomorphism.
    Supplementary problems: Artin, Chapter 2, problems M6 and M7

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