Homomorphism
Disambiguation
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Homomorphism may refer to:
- Homomorphism (category theory) - which all the following are instances of
For types of morphism (eg "epimorphism", "automorphism" and so forth, see:
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
Contents
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
Type | Meaning | Example | Note | Specific example |
---|---|---|---|---|
Endomorphism^{[1]} | 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^{[1]}) | 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^{[1]} | 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
References
- ↑ ^{1.0} ^{1.1} ^{1.2} Algebra - Serge Lang - Revised Third Edition - GTM