A D18: Map $f : G \to H$ is a

**group homomorphism**from $G$ to $H$ if and only if \begin{equation} \forall \, x, y \in G : f(x y) = f(x) f(y) \end{equation}

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Enclosed binary operation

▾ Groupoid

▾ Groupoid homomorphism

▾ Semigroup homomorphism

▾ Monoid homomorphism

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Enclosed binary operation

▾ Groupoid

▾ Groupoid homomorphism

▾ Semigroup homomorphism

▾ Monoid homomorphism

Formulation 0

Let $G$ and $H$ each be a D22: Group.

A D18: Map $f : G \to H$ is a**group homomorphism** from $G$ to $H$ if and only if
\begin{equation}
\forall \, x, y \in G : f(x y) = f(x) f(y)
\end{equation}

A D18: Map $f : G \to H$ is a

Child definitions