11-1100/Homework Assignment 2

#	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 18	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
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See Non Commutative Gaussian Elimination

This assignment is due at class time on Thursday, October 20, 2010.

(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?)
(Selick) Let $H$ be a subgroup of index 2 in a group $G$ . Show that $H$ is normal in $G$ .
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 $σ$ ?
(Selick) Let $G$ be a group of odd order. Show that $x$ is not conjugate to $x - 1$ unless $x = e$ .
(Dummit and Foote) Show that if $G / Z (G)$ is cyclic then $G$ is Abelian.
(Lang) Prove that if the group of automorphisms of a group $G$ is cyclic, then $G$ is Abelian.
(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 $(G:H_1\cap H_2)<\infty$