# group homomorphism

Let $(G,\ast )$ and $(K,\star )$ be two groups. A *group homomorphism ^{}* is a
function $\varphi :G\to K$ such that
$\varphi (s\ast t)=\varphi (s)\star \varphi (t)$ for all $s,t\in G$.

A composition^{} of group homomorphisms is again a homomorphism^{}.

Let $\varphi :G\to K$ a group homomorphism.
Then the kernel of $\varphi $ is a normal subgroup^{} of $G$,
and the image of $\varphi $ is a subgroup^{} of $K$.
Also, $\varphi ({g}^{n})=\varphi {(g)}^{n}$ for all $g\in G$ and for all $n\in \mathbb{Z}$.
In particular,
taking $n=-1$ we have $\varphi ({g}^{-1})=\varphi {(g)}^{-1}$ for all $g\in G$,
and taking $n=0$ we have $\varphi ({1}_{G})={1}_{K}$,
where ${1}_{G}$ and ${1}_{K}$ are the identity elements^{} of $G$ and $K$,
respectively.

Some special homomorphisms have special names.
If the homomorphism $\varphi :G\to K$ is injective^{},
we say that $\varphi $ is a *monomorphism ^{}*,
and if $\varphi $ is surjective

^{}we call it an

*epimorphism*. When $\varphi $ is both injective and surjective (that is, bijective

^{}^{}) we call it an

*isomorphism*. In the latter case we also say that $G$ and $K$ are

^{}*isomorphic*, meaning they are basically the same group (have the same structure

^{}). A homomorphism from $G$ on itself is called an

*endomorphism*, and if it is bijective then it is called an

*automorphism*.

Title | group homomorphism |

Canonical name | GroupHomomorphism |

Date of creation | 2013-05-17 17:53:21 |

Last modified on | 2013-05-17 17:53:21 |

Owner | yark (2760) |

Last modified by | unlord (1) |

Numerical id | 27 |

Author | yark (1) |

Entry type | Definition |

Classification | msc 20A05 |

Synonym | homomorphism |

Synonym | homomorphism of groups |

Related topic | Group |

Related topic | Kernel |

Related topic | Subgroup |

Related topic | TypesOfHomomorphisms |

Related topic | KernelOfAGroupHomomorphism |

Related topic | GroupActionsAndHomomorphisms |

Related topic | Endomorphism2 |

Related topic | GroupsOfRealNumbers |

Related topic | HomomorphicImageOfGroup |

Defines | epimorphism |

Defines | monomorphism |

Defines | automorphism |

Defines | endomorphism |

Defines | isomorphism |

Defines | isomorphic |

Defines | group epimorphism |

Defines | group monomorphism |

Defines | group automorphism |

Defines | group endomorphism^{} |

Defines | group isomorphism |

Defines | epimorphism of groups |

Defines | monomorphism of groups |

Defines | automorphism of a group |

Defines | endom |