HOMOMORPHISMS
Functions that preserve structure. Mapping the rules of one universe onto another without breaking them.
01 // Structure Preservation
A map $\phi$ is a homomorphism if it respects the group operation.
It means: "Do the math, then map" is the same as "Map, then do the math".
φ(a • b)=φ(a) * φ(b)
Operation in G
Operation in H
G
phi (φ)
H
02 // The Anatomy of the Map
The Dead Zone
Kernel (Ker φ)
Everything in G that gets crushed to 0 (the identity) in H. It measures "how much information is lost" during the transfer.
{ g ∈ G | φ(g) = e_H }
The Reach
Image (Im φ)
The actual part of H that gets hit by the map. It might not cover the whole target group.
{ φ(g) | g ∈ G }
Isomorphism
If a homomorphism is a Bijective (one-to-one and onto) map, it is an Isomorphism.
This means the two groups are Identical in every way that matters. They are just labeled differently.
GH