11-1100/Homework Assignment 2

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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