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)
This preview has intentionally blurred sections. Sign up to view the full version.
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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 PALinnell
 Math, Algebra, Group Theory, Normal subgroup, Prime number, Cyclic group, Sylow psubgroup

Click to edit the document details