Homomorphism

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Homomorphism may refer to:

• Notes:Homomorphism - a notes-grade page that may provide some insight.
• Homomorphism (category theory) (AKA: morphisms or arrows of a category. Loosely speaking this is a "structure preserving map", all the homomorphism types listed here are examples of morphism in their respective category.
  • continuous maps - the homomorphisms of topological spaces (not to be confused with homeomorphism) - see also: TOP (category)
  • group homomorphisms
  • linear maps - homomorphisms of vector spaces - see also: WHATEVER THE CATEGORY OF VECTOR SPACES OVER A FIELD IS CALLED!
  • ring homomorphisms

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Flesh out, modules, algebras, measurable spaces!

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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 [math]\text{Hom}(V,W)[/math] is the vector space of all linear maps of the form [math]f:V\rightarrow W[/math], 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:

• [math]\forall a,b\in A[f(a\times_Ab)=f(a)\times_Bf(b)][/math] - 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 [math]f_*:\pi_1(X)\rightarrow\pi_2(X)[/math]) 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:

• A Linear map is a homomorphism between vector spaces