# 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 |
---|---|

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 |