1 Group Theory
1
1
Group Theory
(84 Jan.) Let
G
be a group, with
C
(
G
) its center.
(1) Prove that
C
(
G
) is a normal subgroup of
G
.
(2) Show that the group,
I
(
G
), of all inner automorphisms of
G
is
isomorphic to
G/C
(
G
).
(84 Jan.)
If
H
is a
p
subgroup of a finite group
G
, and
P
is any Sylow
p
subgroup of
G
, then there exists
x
∈
G
so that
H < xPx

1
.
In
particular, any two Sylow
p
subgroups of
G
are conjugate.
(84 Jan.) Prove that the center of a nontrivial finite
p
group contains more
than one element.
(84 Jan.) Let
G
1
and
G
2
be Abelian groups, and let
α
:
G
1
→
G
2
,
β
:
G
2
→
G
1
be group homomorphisms so that
βα
(
g
) =
g
for all
g
∈
G
1
.
(i) Prove that
α
is onetoone and
β
is onto;
(ii) Show that
G
2
=
α
(
G
1
)
⊕
ker
β
.
(84 Aug.)
Let
G
be a finitely generated group.
Prove that every proper
normal subgroup of
G
is included in a normal subgroup of
G
which is
maximal among all proper normal subgroups of
G
.
(84 Aug.)
i) Let
G
be any group. Prove that there is a normal subgroup
N
of
G
such that
G/N
is Abelian and
N
⊆
H
whenever
H
is a normal
subgroup of
G
such that
G/H
is Abelian.
ii) Let
G
be any group and let
Z
(
G
) be the center of
G
. Prove that if
G/Z
(
G
) is cyclic, then
G
is Abelian.
iii) Let
p
be a prime number. Prove that every finite
p
group is solv
able.
(84 Aug.) Describe, up to isomorphism, all the group of order 99.
(84 Aug.)
Let
G
be a finite subgroup of multiplicative group of nonzero
elements of a field. Prove each of the following:
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1 Group Theory
2
i) If
a, b
∈
G
and the orders of
a
and
b
are relatively prime, then the
order of
ab
is
mn
where
m
is the order of
a
and
n
is the order of
b
.
ii) If
a
∈
G
and
k
divides the order of
a
, then there is
c
∈
G
such that
the order of
c
is
k
.
iii) If
a
,
b
∈
G
, then there is
c
∈
G
such that the order of
c
is divisible
by both the order of
a
and the order of
b
.
iv)
G
is cyclic.
(85 Jan.) Let
G
be a group. Prove that
G
cannot have four distinct proper
subgroups
H
0
,
H
1
,
H
2
, and
H
3
such that all of the following conditions
hold:
(i)
H
0
⊆
H
1
⊆
H
2
(ii)
H
0
=
H
2
∩
H
3
(iii)
H
1
H
3
=
G
.
(85 Jan.) Describe, up to isomorphism, all group of order 175.
(85 Jan.)
Let
p
be a prime number.
Suppose
G
is a finite
p
group,
Z
is
the center of
G
, and
N
is nontrivial normal subgroup of
G
. Prove that
N
∩
Z
has more than one element.
(85 Aug.) Let
U
be the multiplicative group of complex numbers of modulus
1. Prove that
U
is isomorphic to the additive group
R
Z
.
(85 Aug.) Suppose that
G
is a group of order
p
n
where
p
is a prime and
H
= 1 is a normal subgroup of
G
. Prove that
Z
(
G
)
∩
H
= 1, where
Z
(
G
) is the center of
G
.
(86 Jan.) Let
C
*
be the multiplicative group of nonzero complex numbers
and let
n
≥
1 be an integer. How many subgroups of
C
*
have order
n
?
What are they? Prove that you have found all of them.
(86 Jan.) If
G
is a group with 30 elements, then prove that
G
has a normal
subgroup
N
with 1
<

N

<
30.
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 Winter '11
 PhanThuongCang
 Group Theory, Normal subgroup, Abelian group

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