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
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
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
is abelian and thus gives a measure of how “close”
G
is to being abelian.)
(a) Prove
G
G
(b) Prove
G/G
is abelian.
(c) If
ϕ
:
G
→
A
is a homomorphism, where
A
is an abelian group, prove that
G
≤
kerϕ
.
Conversely, if
G
≤
kerϕ
, prove that
imϕ
is abelian.
(d) If
G
≤
H
≤
G
, prove that
H
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,
U
(9) = (
{
1
,
2
,
4
,
5
,
7
,
8
}
,
·
(mod 9)). Prove
U
(15)
∼
=
Z
/
4
Z
×
Z
/
2
Z
.
3. Let
Q
be the group of quaternions (see problem 6) and
V
be the Klein4 group.
Prove
Q/Z
(
Q
)
∼
=
V
. (It may help to do problem 6 first)
4. Let
G
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 Spring '11
 Johnson
 Algebra, Abelian group, Sir William Hamilton, commutator subgroup

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