homomorphisms of simple groups
If a group is simple, and is an arbitrary group then any
homomorphism of to must either map all elements of to the
identity
of or be one-to-one.
The kernel of a homomorphism must be a normal subgroup. Since is
simple, there are only two possibilities: either the kernel is all of
of it consists of the identity. In the former case, the
homomorphism will map all elements of to the identity. In the
latter case, we note that a group homomorphism is injective iff the kernel
is trivial.
This is important in the context of representation theory. In that
case, is a linear group and this result may be restated as saying
that representations of a simple group are either trivial or faithful.
Title | homomorphisms of simple groups |
---|---|
Canonical name | HomomorphismsOfSimpleGroups |
Date of creation | 2013-03-22 15:41:59 |
Last modified on | 2013-03-22 15:41:59 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 4 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 20E32 |