# 10-1100/Homework Assignment 2

This assignment is due at class time on Thursday, October 21, 2010.

### Solve the following questions

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