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Math 103A
Final Exam (100 points)
11:30AM Monday 3/19/2007
•
Please put your name and ID number on your blue book.
•
CLOSED BOOK, but BOTH SIDES of two pages of notes are allowed.
•
Calculators are NOT allowed.
•
In a multipart problem, you can do later parts without doing earlier ones.
•
You must show your work to receive credit.
1. (12 pts.) Which are TRUE and which are FALSE? Do NOT give reasons.
(a) If
ϕ
:
G
→
H
is a group homomorphism and
K ⊳ H
,
then
ϕ
−
1
(
K
)
⊳ G
.
(b) If
ϕ
:
G
→
H
is a group homomorphism and
H
is a cyclic group,
then
G
is a cyclic group.
(c) If
H ⊳ G
and
K ⊳ H
, then
K ⊳ G
.
(d) Suppose
p
and
q
are primes.
Then
Z
pq
≈
Z
p
⊕
Z
q
if and only if
p
n
=
q
.
(e) If
G
is a group of order
n
and
k
divides
n
,
then
G
has an element of order
k
.
(f) If
g
∈
G
, then the order of
g
divides the order of
G
.
2. (12 pts.) Let
G
be a group and
Z
(
G
) its center; that is,
Z
(
G
) =
{
g
∈
G

xg
=
gx
for all
x
∈
G
}
.
(a) Prove that
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 Winter '08
 Rogalski,Daniel
 Math, Algebra

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