Math 420/620
Fall 1999
Homework 3
- Given elements , we define the commutator . If , then for all , .
- Consider the group generated by the elements and subject to the relations
This group is denoted and is called the quaternion group.
Show that is a nonabelian group of order 8.
- Consider the following two groups:
- G is generated (multiplicatively) by the real matrices
- H is generated (multiplicatively) by the complex matrices
- Show that G and H are nonabelian groups of order 8.
- Show that precisely one of G and H is isomorphic to
.
- Find all subgroups of symmetric group on three elements.
- We define =
.
- Show that for any group , is itself a group.
- Show that if we define by , then
- Show that the map is a homomorphism from to .
- Let . Compute the number of elements in (
is called an elementary abelian p-group).