Homomorphism

A Homomorphism (not to be confused with homeomorphism) is a structure preserving map.

For example, given vector spaces $V\text{ and }W$ then $$\text{Hom}(V,W)$$ is the vector space of all linear maps of the form $$f:V\rightarrow W$$ , as linear maps will preserve the vector space structure.

Definition

Given two groups $(A,\times_A)$ and $(B,\times_B)$ a map $f:A\rightarrow B$ is a homomorphism if:

• $$\forall a,b\in A[f(a\times_Ab)=f(a)\times_Bf(b)]$$ - note the $\times_A$ and $\times_B$ operations

Note about topological homomorphisms:

Isn't a thing! I've seen 1 book ever (and nothing online) call a continuous map a homomorphism, Homeomorphism is a big thing in topology though. If something in topology (eg $$f_*:\pi_1(X)\rightarrow\pi_2(X)$$ ) it's not talking topologically (as in this case) it's a group (in this case the Fundamental group and just happens to be under the umbrella of Topology

Types of homomorphism

Type	Meaning	Example	Note	Specific example
Endomorphism	A homomorphism from a group into itself	$f:G\rightarrow G$	into doesn't mean injection (obviously)
Isomorphism	A bijective homomorphism	$f:G\rightarrow H$ ($f$ is a bijective)
Monomorphism (Embedding)	An injective homomorphism	$f:G\rightarrow H$ ($f$ is injective)	Same as saying $f:G\rightarrow Im_f(G)$ is an Isomorphism.
Automorphism	A homomorphism from a group to itself	$f:G\rightarrow G$	A surjective endomorphism, an isomorphism from $G$ to $G$	Conjugation

Other uses for homomorphism

The use of the word "homomorphism" pops up a lot. It is not unique to groups. Just frequently associated with. For example:

• A Linear map is a homomorphism between vector spaces