Math 4124
Wednesday, March 30
Second Test Review
The test will cover sections 3.3, 3.5 and chapter 4. Topics will include
1. If
G
is a group,
H
≤
G
and
N
±
G
, then
HN
/
N
∼
=
H
/
H
∩
N
. If furthermore
K
±
G
and
K
≤
N
, then
G
/
K
N
/
K
∼
=
G
/
N
.
2. Transpositions and the alternating group.
A
n
is a normal subgroup of index 2 in
S
n
(
n
≥
2).
3. Groups acting on sets. Orbits and stabilizers.
4. The size of the conjugacy class containing
x
∈
G
is

G
/
C
G
(
x
)

.
5. Left regular representation, conjugation action, conjugacy classes.
6. If

G
/
H

=
n
, then there is a homomorphism
θ
:
G
→
S
n
such that ker
θ
⊆
H
.
7. If
p
is the smallest prime dividing

G

and

G
/
H

=
p
, then
H
±
G
.
8. If

G

=
p
n
where
p
is a prime and
n
≥
1, then Z
(
G
)
6
=
1.
9. If

G

=
p
2
where
p
is a prime, then
G
∼
=
Z
/
p
2
Z
or
G
∼
=
Z
/
p
Z
×
Z
/
p
Z
.
10. Aut
(
G
)
, an automorphism is determined by its affect on a generating set.
11.
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 Spring '08
 Staff
 Math, Algebra, Normal subgroup, Coset, Index of a subgroup, Simple group, Sylow psubgroups

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