Difference between revisions of "Homomorphism"
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For example, given vector spaces {{M|V\text{ and }W}} then <math>\text{Hom}(V,W)</math> is the vector space of all [[Linear map|linear maps]] of the form <math>f:V\rightarrow W</math>, as linear maps will preserve the vector space structure. | For example, given vector spaces {{M|V\text{ and }W}} then <math>\text{Hom}(V,W)</math> is the vector space of all [[Linear map|linear maps]] of the form <math>f:V\rightarrow W</math>, as linear maps will preserve the vector space structure. | ||
| − | == | + | ==Definition== |
Given two [[Group|groups]] {{M|(A,\times)}} and {{M|(B,+)}} a map {{M|f:A\rightarrow B}} is a ''homomorphism'' if: | Given two [[Group|groups]] {{M|(A,\times)}} and {{M|(B,+)}} a map {{M|f:A\rightarrow B}} is a ''homomorphism'' if: | ||
* <math>\forall a,b\in A[f(ab)=f(a)+b(b)]</math> - we need not use different operations (we could use multiplicative for both) but I wanted to emphasise the different groups. | * <math>\forall a,b\in A[f(ab)=f(a)+b(b)]</math> - we need not use different operations (we could use multiplicative for both) but I wanted to emphasise the different groups. | ||
| − | + | ||
| + | '''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]] | 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== | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Type | ||
| + | ! Meaning | ||
| + | ! Example | ||
| + | ! Note | ||
| + | |- | ||
| + | ! Endomorphism<ref name="Lang">Algebra - Serge Lang - Revised Third Edition - GTM</ref> | ||
| + | | A homomorphism from a group ''into'' itself | ||
| + | | {{M|f:G\rightarrow G}} | ||
| + | | ''into'' doesn't mean [[Injection|injection]] (obviously) | ||
| + | |- | ||
| + | ! Automorphism<ref name="Lang"/> | ||
| + | | A homomorphism from a group to itself | ||
| + | | {{M|f:G\rightarrow G}} | ||
| + | | A [[Surjection|surjective]] endomorphism, an isomorphism from {{M|G}} to {{M|G}} | ||
| + | |} | ||
| + | |||
| + | ==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 space|vector spaces]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Category Theory|Abstract Algebra}} | {{Definition|Category Theory|Abstract Algebra}} | ||
Revision as of 20:14, 11 May 2015
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)[/ilmath] and [ilmath](B,+)[/ilmath] a map [ilmath]f:A\rightarrow B[/ilmath] is a homomorphism if:
- [math]\forall a,b\in A[f(ab)=f(a)+b(b)][/math] - we need not use different operations (we could use multiplicative for both) but I wanted to emphasise the different groups.
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 |
|---|---|---|---|
| Endomorphism[1] | A homomorphism from a group into itself | [ilmath]f:G\rightarrow G[/ilmath] | into doesn't mean injection (obviously) |
| 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] |
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
