11-1100/Homework Assignment 2

In Preparation

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

DBN: Classes: 2011-12: 11-1100/Navigation # Week of... Notes and Links Additions to the MAT 1100 web site no longer count towards good deed points 1 Sep 12 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday - NCGE completed, the category of groups, images and kernels. 2 Sep 19 I'll be in Strasbourg, class was be taught by Paul Selick. See my summary. 3 Sep 26 Class Photo, HW1, HW1 Submissions, HW 1 Solutions 4 Oct 3 The Simplicity of the Alternating Groups 5 Oct 10 HW2, HW 2 Solutions 6 Oct 17 Groups of Order 60 and 84 7 Oct 24 Extra office hours: Monday 10:30-12:30 (Dror), 5PM-7PM @ Huron 1028 (Stephen). Term Test on Tuesday. Summary for the midterm exam. Term Test - Sample Solutions 8 Oct 31 HW3, HW 3 Solutions 9 Nov 7 Monday-Tuesday is November Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 14 HW4, HW 4 Solutions 11 Nov 21 12 Nov 28 HW5 and last week's schedule 13 Dec 5 Tuesday - the Jordan form; UofT Fall Semester ends Wednesday; our Final Exam took place on Friday Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

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 $(G:H_{1}\cap H_{2})<\infty$