Syllabus (including office hours and section information)

Midterm Solutions

Final Exam Study Guide

Homework 8 Solutions

A note on homework assignments: reading the text of the sections from which the homework is assigned is implicitly part of the assignment.

Homework 1, due Jan 14, 2010.
Chapter 2 exercises
Section 1 (The Definition of a Group): #1, 4, 5, 10.
Section 2 (Subgroups): #3, 6, 10, 17.
Section 3 (Isomorphisms): #2, 3, 7, 11.

Homework 2, due Jan 21, 2010.
Chapter 2 exercises
Section 4 (Homomorphisms): #5, 12, 13, 8(b).
Section 5 (Equivalence Relations and Partitions): #5, 6, 7.
Section 6 (Cosets): #3, 9, 10.

Homework 3, due Jan 28, 2010.
Chapter 2 exercises
Section 7 (Restriction of a Homomorphism to a Subgroup): #1, 3, 6, 7, 8.
Section 8 (Products of Groups): #4, 8, 9, 10.

Homework 4, due Feb 4, 2010.
Chapter 2 exercises
Section 9 (Modular Arithmetic): #4, 6, 8.
Section 10 (Quotient Groups): #1, 6, 7, 10.
Miscellaneous Problems: #3 (Hint: Partition the group into sets {x, x^{-1}}. Only the elements of order 2 and the identity are singletons, everything else comes in pairs. Now use a parity argument to show that the number of elements of order 2 is odd, and hence nonzero.)

Reminder: There is a midterm on Tuesday, February 9.

Homework 5, due Feb 18, 2010.
Chapter 5 exercises
Section 1 (Symmetry of Plane Figures): #3.
Section 2 (The Group of Motions of the Plane): #7.
Section 3 (Finite Groups of Motions): #2.
Section 5 (Abstract Symmetry: Group Operations): #1, 3, 5, 8, 11(a).

Homework 6, due Feb 25, 2010.
Chapter 5 exercises
Section 6 (The Operation on Cosets): #2, 4, 7.
Section 7 (The Counting Formula): #2, 5.
Section 8 (Permutation Representations): #3, 7, 8.

Homework 7, due Mar 9, 2010.
Chapter 5 exercises
Section 9 (Finite Subgroups of the Rotation Group): #1, 3, 5.
Chapter 6 exercises
Section 1 (The Operations of Group on Itself): #2, 4, 7, 8(e), (f).
Section 2 (The Class Equation of the Icosahedral Group): #6, 9.

Homework 8, due Mar 11, 2010.
Chapter 6 exercises
Section 3 (Operations on Subsets): #1, 6, 13, 14, 15. Hint for #14: How many conjugate subgroups does H have? Prove this is less than or equal to [G:H]. How many elements does each conjugate subgroup have? So all together, how many elements does this account for? Now use the fact that there is overlap, since every subgroup contains the identity.

Section 4 (The Sylow Theorems): #1, 5, 9, 16. There are two ways I can see to do #9. One is to replicate the proof of the First Sylow Theorem in the book, with p^e replaced by p^r. The second way is to apply the First Sylow Theorem (and induction) to reduce the problem to the statement: "Every group of order p^e with e>=1 contains a subgroup of order p^(e-1)." To prove this statement, first prove the lemma that every p-group of size p^e with e>=2 has a normal subgroup H of size p^s for some s, with 1<= s <= e-1. Once you have the lemma, you can use induction to prove the statement in quotes. Indeed, G/H is a p-group of size p^(e-s). So G/H has a subgroup K of size p^(e-s-1) by induction. Then the inverse image of K in G (i.e. the set {x in G such that the coset xH is in K}) is a subgroup of G of size p^(e-1). Now how do you prove the lemma? Well, if G is abelian, then letting H be the subgroup generated by any element of order p does the trick. If G is not abelian, then letting H be the center of G does the trick.