Problem 2.3: Which of the following collections of 2 2 matrices with real
entries form groups under matrix multiplication?
ab
i) Those of the form
for which ac = b2
bd
Answer: The set of such matrices
Midterm solutions.
Problem 1.
(1) Dene what it means for two groups to be isomorphic.
(2) Dene the order of an element of a group. Give an example to show that
the order can be innite. No proof is nec
Homework 8 solutions.
Problem 16.1. Which of the following dene homomomorphisms from C \ cfw_0 to
C \ cfw_0?
Answer. a) f1 : z z
Yes, f1 is a homomorphism. We have that z is the complex conjugate of
Homework 6 solutions.
Problem 14.1. Work out the conjugacy classes of D5 .
Answer. We have that
D5 = cfw_e, r, r2 , r4 , s, sr, sr2 , sr3 , sr4
The conjugacy class of e is just cfw_e. To nd the conju
Homework 6 solutions.
Problem 9.1. Which of the following collections of n n real matrices form groups
under matrix multiplication?
Proof. In each of the following cases, we will use the following not
Homework 4 solutions.
Problem 11.2. Let H be a subgroup of a group G. Prove that g1 H = g2 H if and
only if g1 1 g2 belongs to H .
Proof. Suppose g1 H = g2 H . Since H is a subgroup, the identity e is
Homework 4 solutions.
Problem 7.4. Produce a specic isomorphism between S3 and D3 . How many
dierent isomorphisms are there from S3 to D3 ?
Answer. S3 = cfw_e, (12), (13), (23), (123), (132) and D3 =
Homework 3 solutions.
Problem 6.1. Write out a multiplication table for S3 .
Answer.
e
(12)
(13)
(23)
(123)
(132)
e
e
(12)
(13)
(23)
(123)
(132)
(12)
(12)
e
(123)
(132)
(13)
(23)
(13)
(13)
(132)
e
(12
Homework 2 solutions.
Problem 4.4. Let g be an element of the group G. Keep g xed and let x vary
through G. Prove that the products gx are all distinct and ll out G. Do the same
for the products xg .
Revised version: The rst version had a few questions that were a bit too hard.
Notation: Zm denotes the integers modulo m under addition.
(1) Give examples of each of the following. No proof or justic