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Math 5125
Wednesday, September 28
First Test. Answer All Problems.
Please Give Explanations For Your Answers
1. Let
H
be a subgroup of the ﬁnite group
G
, and let
p
be a prime. Prove that two
distinct Sylow
p
subgroups of
H
cannot be contained in the same
p
subgroup of
G
.
(16 points)
2. Prove that a group of order 280 cannot be simple.
(17 points)
3. Let
A
and
B
be ﬁnitely generated abelian groups. Prove that if
A
×
A
∼
=
B
×
B
, then
A
∼
=
B
.
(17 points)
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View Full DocumentMath 5125
Wednesday, September 28
Solutions to First Test
1. Let
P
1
,
P
2
be Sylow
p
subgroups of
H
, and suppose
Q
is a
p
subgroup of
G
containing
P
1
and
P
2
. Since
Q
∩
H
is a
p
group containing
P
1
and
P
1
is a Sylow
p
subgroup, we
must have
Q
∩
H
=
P
1
. Similarly
Q
∩
H
=
P
2
and we conclude that
P
1
=
P
2
, as required.
2. Let
G
be a simple group of order 280
=
8
*
5
*
7. The number of Sylow 2subgroups is
congruent to 1 mod 7 and divides 40. This number cannot be 1, because then
G
would
have a normal subgroup of order 8. Therefore
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 Fall '07
 PALinnell
 Math, Algebra

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