Homomorphism
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 |
TODO: Make sure these definitions are in the same order (and all here) compared to Linear map
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
