Math 720

Fall 2000

Exam 1

For this exam, p and q will always refer to nonzero prime integers. For the first two problems you may assume that p and q are both odd primes.

1. Classify all groups of order pq.

2. Classify all groups of order p³.

3. Let with Show that the number of Sylow p-subgroups divides m.

4. Show that if has precisely two conjugates, then is not simple.

5. Let be a group containing an element of order greater than 2, show that is nontrivial. (Extra credit: show if then is nontrivial).

6. Let be a group of order p²q² such that p>q². Show that is the semidirect product of two abelian groups.

7. Let and assume that G is simple. How many elements of order 7 must G possess?