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 is not closed under matrix multiplication, so
it does
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 necessary.
Proof.
(1) Two groups G and H are isomorphic if
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
z . If z1 , z2 are two complex numbers, then (z1 z2 ) =
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 conjugacy class of rn , note that
when we conjugate rn by rm
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 notation. If A is
an n n matrix, we write A = (ai,j ) wher
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 in H . So
g2 g2 H . Since g1 H = g2 H , there is some
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 = cfw_e, r, r2 , s, sr, sr2 . The
group D3 is the group o
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
(123)
(23)
(12)
(23)
(23)
(123)
(132)
e
(12)
(123)
(123)
(
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 .
Proof. Let g G. Let x1 = x2 G. We need to show that gx1
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 justication is necessary:
(a) A group which is not cyclic;
(b