11-1100/Homework Assignment 1

In Preparation

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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
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See Non Commutative Gaussian Elimination

This assignment is due at class time on Tuesday, October 11, 2011.

Part I

Web search "Rubik's Cube Variants" (look at images), or look at Wikipedia: Combination Puzzle or TwistyPuzzles.com, or search elsewhere or go to a toy shop, pick your favourite "permutation group puzzle" (other than the Rubik Cube, of course), and figure out how many configurations it has. For your solution to count the number of configurations must be more than you can count, and your solution must include a clear picture or diagram of the object being studied, its labeling by integers, the list of generating permutations for it, and a printout of the program you used along with screen shot of its output (or an input/output log). It is ok to use the program presented in class (Mathematica is available on a departmental server; look for it!) but better to write your own. You can submit your solution either as a wiki page on this server (best option), or as a URL elsewhere (second best), or as a single file in any reasonable format, or on paper.

Part II

Solve the following questions.

(Selick) If $g$ is an element of a group $G$ , the order $|g|$ of $g$ is the least positive number n for which $g^{n}=1$ (may be $\infty$ ). If $x,y\in G$ , prove that $|xy|=|yx|$ .
(Selick) Let $G$ be a group. Show that the function $\phi :G\to G$ given by $\phi (g)=g^{2}$ is a morphism of groups if and only if $G$ is Abelian.
(Lang, pp 75) Let $G$ be a group. For $a,b\in G$ , the commutator $[a,b]$ of $a$ and $b$ is $[a,b]=aba^{-1}b^{-1}$ . Let $G'$ be the subgroup of $G$ generated by all commutators of elements of $G$ . Show that $G'$ is normal in $G$ , that $G/G'$ is Abelian, and that any morphism from $G$ into an Abelian group factors through $G/G'$ .
(Lang, pp 75) Let $G$ be a group. An automorphism of $G$ is an invertible group morphism $G\to G$ . An inner automorphism is an automorphism of $G$ given by conjugation by some specific element $g$ of $G$ , so $x\mapsto x^{g}$ . Prove that the inner automorphisms of $G$ form a normal subgroup of the group of all automorphisms of $G$ .