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Math 4124
Monday, March 14
Sample Second Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let
G
be a ﬁnite group, let
H
≤
G
, and let
N
±
G
. If
(

N

,

G
/
H

) =
1, prove that
N
⊆
H
.
(10 points)
2. Let
G
=
S
4
and let
π
:
G
→
S
G
denote the regular representation of
G
(so
S
G
∼
=
S
24
).
If
t
∈
G
is a transposition, what is the cycle decomposition of
π
t
? Use this to show
that
G
is isomorphic to a subgroup of
A
24
.
(10 points)
3. Compute the orders of the conjugacy class and the centralizer of (1 2 3)(4 5 6)(7 8) in
S
10
.
(10 points)
4. Let
p
be a prime and let
G
be a nonabelian group of order
p
4
. Prove that

Z
(
G
)

=
p
or
p
2
.
(10 points)
5. Show that
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This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.
 Spring '08
 Staff
 Math, Algebra

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