# group homomorphism

Let $(G,\ast)$ and $(K,\star)$ be two groups. A group homomorphism is a function $\phi\colon G\to K$ such that $\phi(s\ast t)=\phi(s)\star\phi(t)$ for all $s,t\in G$.

A composition of group homomorphisms is again a homomorphism.

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

Some special homomorphisms have special names. If the homomorphism $\phi\colon G\to K$ is injective, we say that $\phi$ is a monomorphism, and if $\phi$ is surjective we call it an epimorphism. When $\phi$ 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