homomorphisms of simple groups
If a group G is simple, and H is an arbitrary group then any homomorphism of G to H must either map all elements of G to the identity of H or be one-to-one.
The kernel of a homomorphism must be a normal subgroup. Since G is simple, there are only two possibilities: either the kernel is all of G of it consists of the identity. In the former case, the homomorphism will map all elements of G 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, H 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 |
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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 |