Math 420/620
Exam 1
Fall 1999
- (Fermat’s Little Theorem) Prove that if
is a nonzero prime number and
is any integer, then
.
- Let
be a group and define the mapping
by ![](Image498.gif)
.
- Show that
is an automorphism of
if and only if
is abelian.
- Use part a) to show that if
is a finite abelian group and
then ![](Image503.gif)
is a finite group of even order.
- Let
. Prove that
.
- Assume that
is a group of finite order and
is a normal subgroup of ![](Image455.gif)
.
- If
is a subgroup of
with
, then
.
- Use part a) to show that if
, then
is the unique subgroup of
of order ![](Image514.gif)
- (G)
Classify all groups of order 2p with p a prime integer.
6. (G) Find a formula expressing the number of distinct abelian groups of order
where
is a prime number.