Latest revision as of 22:04, 19 October 2016

Disambiguation

This page lists articles associated with the same title.

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Homomorphism may refer to:

• Homomorphism (category theory) - which all the following are instances of
  • Homomorphism (group)
  • Homomorphism (module)
  • Homomorphism (ring)
  • Homomorphism (topology) - AKA: continuous map
  • Homomorphism (vector space) - AKA: linear map - instance of a module homomorphism

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

• Types of homomorphism

OLD STUFF

• 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

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 [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

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

References

1. ↑ ^{1.0 1.1 1.2} Algebra - Serge Lang - Revised Third Edition - GTM