Math 420/620

Homework 5

Fall 1999

 

  1. If has finite index in the group , then show that there is a normal subgroup , with and .

    2.  

  2. Classify all groups of order with a nonzero prime number.

    3.  

  3. Let be distinct odd primes with
  1. Assume that does not divide , classify all groups of order
  2. (G) Classify all groups of order
without the above assumption.

 

  1. Let be a group of order for a nonzero prime . Prove that has subgroups of or der for all

    2.  

  2. How many elements of order 7 must a simple group of order 168 have?

    3.  

  3. If , 2907, or 6545, show that is not simple.

    4.  

  4. (G) Let with distinct primes. Show is not simple.

    5.  

  5. (G) Show if is a nonabelian simple group of order less than 100, then
.