Simple explanation of homomorphisms?

Question

Could someone please explain homomorphisms to me in a simple way? I've heard the following definition:

"Let $G,H$ be groups. A mapping $\theta : G \rightarrow H$ is a homomorphism iff $\theta (xy) = \theta (x) \theta (y)$ for all $x,y \in G$"

However, I'm struggling to understand the actual meaning of this statement.

What is the geometric interpretation of $\theta (xy) = \theta (x) \theta (y)$? Does this mean that it is a linear mapping? If not, how is such a mapping different to a mapping which is not homomorphic?

Possible duplicated here: http://math.stackexchange.com/questions/242348/intuition-on-group-homomorphisms — SiXUlm, Nov 15 '15 at 14:14
Very clear video: https://www.youtube.com/watch?v=cYzp5IWqCsg — D.R., Dec 14 '19 at 06:51

score 0 · Answer 1 · answered Nov 15 '15 at 14:50

Roughly speaking, a homomorphism is a map which respects the underlying structure of the set. In the case of a group it means the map changes a product into a product.

If the notation is additive, it changes a sum into a sum.

A well-known example will hopefully shed some light: logarithm functions are defined on the multiplicative group $\mathbf R^{*+}$, have range the additive group $\mathbf R$ and their fundamental properties are $$\begin{cases} \log(xy)=\log x+\log y,\\ \log 1=0. \end{cases}$$ Hence logarithms are group homomorphisms (an even (continuous) group isomorphisms).

Simple explanation of homomorphisms?

1 Answers1