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group homomorphism (Definition)

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.



"group homomorphism" is owned by yark. [ full author list (4) | owner history (2) ]
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See Also: group, kernel, subgroup, types of homomorphisms, kernel, group actions and homomorphisms, endomorphism, the groups of real numbers

Other names:  homomorphism, homomorphism of groups
Also defines:  epimorphism, monomorphism, automorphism, endomorphism, isomorphism, isomorphic, group epimorphism, group monomorphism, group automorphism, group endomorphism, group isomorphism, epimorphism of groups, monomorphism of groups, automorphism of a group, endomorphism of a group, isomorphism of groups
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Cross-references: bijective, surjective, injective, identity elements, subgroup, image, normal subgroup, kernel, composition, function, groups
There are 234 references to this entry.

This is version 22 of group homomorphism, born on 2001-11-08, modified 2006-10-16.
Object id is 719, canonical name is GroupHomomorphism.
Accessed 27919 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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