# 11-1100/Homework Assignment 2

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 Sn contains an element of order 18?
2. What is the maximal order of an element in S26? (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 σ?
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 H1 and H2 be subgroups of G. Suppose $(G:H_1)<\infty$ and $(G:H_2)<\infty$. Show that $(G:H_1\cap H_2)<\infty$