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
![](Image440.gif)
![](Image441.gif)
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
![](Image444.gif)
- H is generated (multiplicatively) by the complex matrices
![](Image445.gif)
- Show that G and H are nonabelian groups of order 8.
- Show that precisely one of G and H is isomorphic to
![](Image446.gif)
.
- Find all subgroups of symmetric group on three elements.
- We define
=![](Image448.gif)
.
- Show that for any group
,
is itself a group.
- Show that if we define
by
, then ![](Image453.gif)
- Show that the map
is a homomorphism from
to
.
- Let
. Compute the number of elements in
(![](Image459.gif)
is called an elementary abelian p-group).