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
.
- 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
is a finite group of even order.
- Let . Prove that .
- Assume that is a group of finite order and is a normal subgroup of
.
- If is a subgroup of with , then .
- Use part a) to show that if , then is the unique subgroup of of order
- (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.