group homomorphism GroupHomomorphism 2001-11-08 16:24:02 2006-10-16 17:14:06 Definition 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 \usepackage{amsfonts} \def\Z{\mathbb{Z}} \PMlinkescapeword{endomorphism} \PMlinkescapeword{structure} Let $(G,\ast)$ and $(K,\star)$ be two groups. A \emph{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 $n \in\Z$. 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 \emph{monomorphism}, and if $\phi$ is surjective we call it an \emph{epimorphism}. When $\phi$ is both injective and surjective (that is, bijective) we call it an \emph{isomorphism}. In the latter case we also say that $G$ and $K$ are \emph{isomorphic}, meaning they are basically the same group (have the same structure). A homomorphism from $G$ on itself is called an \emph{endomorphism}, and if it is bijective then it is called an \emph{automorphism}.