1. (Fermat’s Little Theorem) Prove that if is a nonzero prime number and is any integer, then .

 

2. Let be a group and define the mapping by

1. Show that is an automorphism of if and only if is abelian.
2. Use part a) to show that if is a finite abelian group and then

1. Let . Prove that .

 

2. Assume that is a group of finite order and is a normal subgroup of

1. If is a subgroup of with , then .
2. Use part a) to show that if , then is the unique subgroup of of order

1. (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.