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MATH 4107
EXAM #2
March 19, 2010
PLEASE READ THESE DIRECTIONS: Answer PROBLEM 1 (20 points) and choose TWO
other problems to answer (15 points each). You may also answer (for up to 5 points extra credit)
ONE additional problem. In this case, please specify which problem is the extra credit problem.
All statements require proof or justiFcation. There are 50 points total, plus up to 5 points of
extra credit.
1. Let
A
,
B
be subgroups of a Fnite group
G.
(a) Let
AB
=
{
ab
:
a
∈
A,b
∈
B
}
.
Prove that if
A
∩
B
=
{
e
}
then

AB

=

A

B

.
(b) Show that if
A
is a
normal
subgroup of
G
then
AB
is closed under compositions
G.
Remark: In fact,
AB
is a subgroup in this case, but you only have to prove that it is closed
under compositions.
(c) Show that if
A
and
B
are
both normal
then
AB
is normal in
G.
Note: You just have to prove that it is normal, the subgroup part is covered by part (b).
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This note was uploaded on 08/25/2011 for the course MATH 4107 taught by Professor Staff during the Spring '10 term at University of Florida.
 Spring '10
 Staff
 Algebra, Addition

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