Math 420/620

Fall 1999

Homework 3

 

  1. Given elements , we define the commutator . If , then for all , .

    2.  

  2. Consider the group generated by the elements and subject to the relations

    3. This group is denoted and is called the quaternion group.

    Show that is a nonabelian group of order 8.

     

  3. Consider the following two groups:

  1. Show that G and H are nonabelian groups of order 8.
  2. Show that precisely one of G and H is isomorphic to
.

 

 

  1. Find all subgroups of symmetric group on three elements.

    2.  

  2. We define =
.
  1. Show that for any group , is itself a group.
  2. Show that if we define by , then
  3. Show that the map is a homomorphism from to .
  4. Let . Compute the number of elements in (
is called an elementary abelian p-group).