Homomorphisms

Overview

A homomorphism is a map $\varphi$ between two algebraic structures $S$ and $T$, of the same type, that preserves the operations of the structures. That is, for any operation $\mu$ on $S$ and corresponding operation ${\mu }^{\prime }$ on $T$,

The above identity is called the homomorphism property.

Homomorphisms Set

Let $S$ and $T$ be algebraic structures of the same kind. The set of homomorphisms from $S$ to $T$ is denoted $\text{Hom}(S,T)$.

Endomorphisms

An endomorphism is a homomorphism from an algebraic structure to itself.

Isomorphisms

Let $S$ and $S^{\prime }$ be algebraic structures of the same kind. An isomorphism is a bijective function $\varphi :S\to S^{\prime }$ satisfying the homomorphism property. If such an isomorphism exists, then we say that $S$ is isomorphic to $S^{\prime }$, denoted $S\simeq S^{\prime }$.

Automorphisms

An automorphism is an isomorphism from an algebraic structure to itself. The trivial automorphism is simply the identity function.

The automorphism group of an object $X$ is the group consisting of automorphisms of $X$ under composition, denoted $\text{Aut}(X)$.

• If $X$ is a set, then $\text{Aut}(X)$ corresponds to the permutation group $\text{Sym}(X)$ of $X$.
• If $G$ is a group, then $\text{Aut}(G)$ is a subgroup of $\text{Sym}(G)$.

Composition

Let $\varphi :A\to B$ and $\psi :B\to C$ be homomorphisms. Then $\psi \circ \varphi$ is also a homomorphism.

Special Cases

Injection Maps

Let $S=S_1\times S_2\times \cdots \times S_n$ be the direct product of same-typed algebraic structures. If $e_i\in S_i$ is the identity element for each $i=1,2,\dots ,n$, the injection map ${\varphi }_i:S_i\to S$ given by

is a homomorphism for each $i=1,2,\dots ,n$.

Projection Maps

Let $S=S_1\times S_2\times \cdots \times S_n$ be the direct product of same-typed algebraic structures. The projection map ${\pi }_i:S\to S_i$ given by

is a homomorphism for each $i=1,2,\dots ,n$.