10-1100/Homework Assignment 1

DBN: Classes: 2010-11: 10-1100/Navigation Additions to the MAT 1100 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 13 About This Class, Tuesday - Non Commutative Gaussian Elimination, Thursday 2 Sep 20 Tuesday - Homomorphisms and Normal Groups, Thursday - Isomorphism Theorems 3 Sep 27 Class Photo, HW1, HW1 solution 4 Oct 4 5 Oct 11 HW2, HW2 solution 6 Oct 18 7 Oct 25 Term Test 8 Nov 1 HW3, HW3 solution 9 Nov 8 Monday-Tuesday is Fall Break, One Theorem, Two Corollaries, Four Weeks 10 Nov 15 HW4, HW4 solution 11 Nov 22 12 Nov 29 HW5, HW5 solution 13 Dec 6 Boxing Day Handout, see also December 2010 Schedule. Some Class Notes F Dec 13 Final exam, Tuesday December 14 10-1, Bahen 6183 Register of Good Deeds Add your name / see who's in! See Non Commutative Gaussian Elimination

In Preparation

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

This assignment is due at class time on Tuesday, October 12, 2010.

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.

1. (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 $|xy|=|yx|$.
2. (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.
3. (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 $G/G'$.
4. (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 $G$.