Homework 6 Solutions
Math 332, Spring 2010
Problem 1.
(a)
Proposition.
If
G
and
H
are groups, then
G
×
H
≈
H
×
G
.
Proof.
Deﬁne
ϕ
:
G
×
H
→
H
×
G
by
ϕ
(
g,h
) = (
h,g
). Clearly
ϕ
is bijective. Moreover, if
(
g
1
,h
1
)
,
(
g
2
,h
2
)
∈
G
×
H
, then
ϕ
(
(
g
1
,h
1
)(
g
2
,h
2
)
)
=
ϕ
(
g
1
g
2
,h
1
h
2
) = (
h
1
h
2
,g
1
g
2
) = (
h
1
,g
1
)(
h
2
,g
2
) =
ϕ
(
g
1
,h
1
)
ϕ
(
g
2
,h
2
)
,
which proves that
ϕ
is an isomorphism.
(b)
Proposition.
Let
G
and
H
be groups. If
A
≤
G
and
B
≤
H
, then
A
×
B
≤
G
×
H
.
Proof.
We shall use the onestep subgroup test. Clearly
A
×
B
is a nonempty subset of
G
×
H
.
Furthermore, if (
a
1
,b
1
)
,
(
a
2
,b
2
)
∈
A
×
B
, then
a
1
,a
2
∈
A
and
b
1
,b
2
∈
B
. Since
A
and
B
are
subgroups, it follows that
a
1
a

1
2
∈
A
and
b
1
b

1
2
∈
B
, so
(
a
1
,b
1
)(
a
2
,b
2
)

1
= (
a
1
,b
1
)(
a

1
2
,b

1
2
) = (
a
1
a

1
2
,b
1
b

1
2
)
∈
A
×
B.
(c)