MATH 4107
PRACTICE PROBLEMS WITH SOLUTIONS
March 2, 2007
Here are a few practice problems on groups.
Try to work these WITHOUT looking at
the solutions!
After you write your own solution, you can compare to my solution.
Your
solution does not need to be identical—there are often many ways to solve a problem—but
it does need to be CORRECT.
1.
Suppose
G
,
A
, and
B
are groups and
ψ
1
:
G
→
A
and
ψ
2
:
G
→
B
are surjective
homomorphisms. Suppose, moreover that ker(
ψ
1
)
∩
ker(
ψ
2
) =
{
e
}
.
a. Show that
ψ
:
G
→
A
×
B
, deFned by
ψ
(
g
) = (
ψ
1
(
g
)
, ψ
2
(
g
)) is an injective homomor
phism of
G
into
A
×
B
.
Solution
If
g
,
h
∈
G
, then, since
ψ
1
,
ψ
2
are both homomorphisms,
ψ
(
gh
)
=
(
ψ
1
(
gh
)
, ψ
2
(
gh
)
)
=
(
ψ
1
(
g
)
ψ
1
(
h
)
, ψ
2
(
g
)
ψ
2
(
h
)
)
=
(
ψ
1
(
g
)
, ψ
2
(
g
)
) (
ψ
1
(
h
)
, ψ
2
(
h
)
)
=
ψ
(
g
)
ψ
(
h
)
.
Thus
ψ
is a homomorphism.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Staff
 Math, Algebra, Algebraic structure

Click to edit the document details