This preview shows page 1. Sign up to view the full content.
Homework 5
Due: Friday 2/26/10
Problems after the double bar will not be collected but it is recommended that you do them. My
assumption is that you will do them.
1. The
commutator subgroup
G
0
of a group
G
is the subgroup generated by the set
{
x

1
y

1
xy
:
x,y
∈
G
}
. That is, every element of
G
0
has the form
a
i
1
1
a
i
2
2
···
a
i
k
k
where each
a
j
has the form
x

1
y

1
xy
, each
i
j
=
±
1 and
k
is any positive integer. (The commutator subgroup is the
smallest normal subgroup such that
G/G
0
is abelian and thus gives a measure of how “close”
G
is to being abelian.)
(a) Prove
G
0
C
G
(b) Prove
G/G
0
is abelian.
(c) If
ϕ
:
G
→
A
is a homomorphism, where
A
is an abelian group, prove that
G
0
≤
kerϕ
.
Conversely, if
G
0
≤
kerϕ
, prove that
imϕ
is abelian.
(d) If
G
0
≤
H
≤
G
, prove that
H
C
G
.
2. For
n
a positive integer,
U
(
n
) is the group consisting of the positive integers less than
n
and
relatively prime to
n
with the binary operation of multiplication modulo
n
. For example,
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '11
 Johnson
 Algebra

Click to edit the document details