# Homomorphism

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## Disambiguation

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Homomorphism may refer to:

For types of morphism (eg "epimorphism", "automorphism" and so forth, see:

# OLD STUFF

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Flesh out, modules, algebras, measurable spaces!

# OLD PAGE

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

For example, given vector spaces [ilmath]V\text{ and }W[/ilmath] 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 [ilmath](A,\times_A)[/ilmath] and [ilmath](B,\times_B)[/ilmath] a map [ilmath]f:A\rightarrow B[/ilmath] is a homomorphism if:

• $\forall a,b\in A[f(a\times_Ab)=f(a)\times_Bf(b)]$ - note the [ilmath]\times_A[/ilmath] and [ilmath]\times_B[/ilmath] 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 [ilmath]f:G\rightarrow G[/ilmath] into doesn't mean injection (obviously)
Isomorphism A bijective homomorphism [ilmath]f:G\rightarrow H[/ilmath] ([ilmath]f[/ilmath] is a bijective)
Monomorphism (Embedding) An injective homomorphism [ilmath]f:G\rightarrow H[/ilmath] ([ilmath]f[/ilmath] is injective) Same as saying [ilmath]f:G\rightarrow Im_f(G)[/ilmath] is an Isomorphism.
Automorphism A homomorphism from a group to itself [ilmath]f:G\rightarrow G[/ilmath] A surjective endomorphism, an isomorphism from [ilmath]G[/ilmath] to [ilmath]G[/ilmath] 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: