Definition Group homomorphism

Group homomorphisms are mappings between two groups that are compatible with the group structure. Similarly, field homomorphisms and vector space homomorphisms will be considered later.

Let $(G, \circ_G)$ and $(H, \circ_H)$ be groups. A \textit{group homomorphism} is a mapping $f : G \to H$ such that

\[

f(g \circ_G g') = f(g) \circ_H f(g')

\]

for all $g, g' \in G$.