# 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 $(G:H_1\cap H_2)<\infty$