Difference between revisions of "Homomorphism"
(Created page with "A Homomorphism '''(not to be confused with homeomorphism)''' is a structure preserving map. For example, given vector spaces {{M|V\text{ and }W}} then <math...") |
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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|Category Theory}} | + | ==Group homomorphism== |
| + | 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. | ||
| + | |||
| + | ==Topological homomorphism== | ||
| + | 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]] | ||
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| + | {{Definition|Category Theory|Abstract Algebra}} | ||
Revision as of 15:05, 16 April 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.
Group homomorphism
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.
Topological homomorphism
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
