Math 420/620

Exam 1

Fall 1999

 

  1. (Fermat’s Little Theorem) Prove that if is a nonzero prime number and is any integer, then .
  2.  

  3. 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
is a finite group of even order.

 

  1. Let . Prove that .
  2.  

  3. 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.