11-1100/Homework Assignment 2

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In Preparation

The information below is preliminary and cannot be trusted! (v)

This assignment is due at class time on Thursday, October 20, 2010.

Solve the following questions

  1. (Selick)
    1. What it the least integer n for which the symmetric group S_n contains an element of order 18?
    2. What is the maximal order of an element in S_{26}? (That is, of a shuffling of the red cards within a deck of cards?)
  2. (Selick) Let H be a subgroup of index 2 in a group G. Show that H is normal in G.
  3. Let \sigma\in S_{20} be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer C_{S_{20}}(\sigma) of \sigma?
  4. (Selick) Let G be a group of odd order. Show that x is not conjugate to x^{-1} unless x=e.
  5. (Dummit and Foote) Show that if G/Z(G) is cyclic then G is Abelian.
  6. (Lang) Prove that if the group of automorphisms of a group G is cyclic, then G is Abelian.
  7. (Lang)
    1. Let G be a group and let H be a subgroup of finite index. Prove that there is a normal subgroup N of G, contained in H, so that (G:N) is also finite. (Hint: Let (G:H)=n and find a morphism G\to S_n whose kernel is contained in H.)
    2. Let G be a group and H_1 and H_2 be subgroups of G. Suppose (G:H_1)<\infty and (G:H_2)<\infty. Show that (G:H_1\cap H_2)<\infty