Difference between revisions of "Homomorphism"
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| + | {{Refactor notice|grade=B}} | ||
| + | {{Disambiguation}} | ||
| + | * [[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}}: [[morphism|morphisms]] or [[arrow|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. | ||
| + | ** {{plural|continuous map|s}} - the homomorphisms of {{plural|topological space|s}} (not to be confused with [[homeomorphism]]) - see also: [[TOP (category)]] | ||
| + | ** {{plural|group homomorphism|s}} | ||
| + | ** {{plural|linear map|s}} - homomorphisms of {{plural|vector space|s}} - see also: WHATEVER THE CATEGORY OF VECTOR SPACES OVER A FIELD IS CALLED! | ||
| + | ** {{plural|ring homomorphism|s}} | ||
| + | |||
| + | {{Stub page|grade=B|msg=Flesh out, modules, algebras, measurable spaces!}} | ||
| + | {{Definition|Category Theory|Linear Algebra|Topology}} | ||
| + | |||
| + | |||
| + | <hr/> | ||
| + | |||
| + | |||
| + | __TOC__ | ||
| + | =OLD PAGE= | ||
A Homomorphism '''(not to be confused with [[Homeomorphism|homeomorphism]])''' is a structure preserving map. | A Homomorphism '''(not to be confused with [[Homeomorphism|homeomorphism]])''' is a structure preserving map. | ||
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==Definition== | ==Definition== | ||
| − | Given two [[Group|groups]] {{M|(A,\ | + | Given two [[Group|groups]] {{M|(A,\times_A)}} and {{M|(B,\times_B)}} a map {{M|f:A\rightarrow B}} is a ''homomorphism'' if: |
| − | * <math>\forall a,b\in A[f( | + | * <math>\forall a,b\in A[f(a\times_Ab)=f(a)\times_Bf(b)]</math> - note the {{M|\times_A}} and {{M|\times_B}} operations |
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! Example | ! Example | ||
! Note | ! Note | ||
| + | ! Specific<br/>example | ||
|- | |- | ||
! Endomorphism<ref name="Lang">Algebra - Serge Lang - Revised Third Edition - GTM</ref> | ! Endomorphism<ref name="Lang">Algebra - Serge Lang - Revised Third Edition - GTM</ref> | ||
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| {{M|f:G\rightarrow G}} | | {{M|f:G\rightarrow G}} | ||
| ''into'' doesn't mean [[Injection|injection]] (obviously) | | ''into'' doesn't mean [[Injection|injection]] (obviously) | ||
| + | | | ||
|- | |- | ||
! Isomorphism | ! Isomorphism | ||
| A [[Bijection|bijective]] homomorphism | | A [[Bijection|bijective]] homomorphism | ||
| {{M|f:G\rightarrow H}} ({{M|f}} is a [[Bijection|bijective]]) | | {{M|f:G\rightarrow H}} ({{M|f}} is a [[Bijection|bijective]]) | ||
| + | | | ||
| | | | ||
|- | |- | ||
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| {{M|f:G\rightarrow H}} ({{M|f}} is [[Injection|injective]]) | | {{M|f:G\rightarrow H}} ({{M|f}} is [[Injection|injective]]) | ||
| Same as saying {{M|f:G\rightarrow Im_f(G)}} is an [[Isomorphism]]. | | Same as saying {{M|f:G\rightarrow Im_f(G)}} is an [[Isomorphism]]. | ||
| + | | | ||
|- | |- | ||
! Automorphism<ref name="Lang"/> | ! Automorphism<ref name="Lang"/> | ||
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| {{M|f:G\rightarrow G}} | | {{M|f:G\rightarrow G}} | ||
| A [[Surjection|surjective]] endomorphism, an isomorphism from {{M|G}} to {{M|G}} | | A [[Surjection|surjective]] endomorphism, an isomorphism from {{M|G}} to {{M|G}} | ||
| + | | [[Conjugation]] | ||
|} | |} | ||
| − | |||
==Other uses for homomorphism== | ==Other uses for homomorphism== | ||
Latest revision as of 22:04, 19 October 2016
Grade: B
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Disambiguation
This page lists articles associated with the same title.
If an internal link led you here, you may wish to change the link to point directly to the intended article.
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
Stub grade: B
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Flesh out, modules, algebras, measurable spaces!
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
