= Group homomorphism
{title2=$\varphi(gh)=\varphi(g)\varphi(h)$}
{wiki}
A <group homomorphism> is a <map> $\varphi:G\to H$ preserving the <group operation>: $\varphi(gh)=\varphi(g)\varphi(h)$. It also preserves the <identity element> and <inverse elements>. In an additive <group>, the same requirement reads $\varphi(g+h)=\varphi(g)+\varphi(h)$. A <group homomorphism> preserves a way of combining inputs, even when several inputs have the same output.
Reduction modulo three from $\mathbb Z/6\mathbb Z$ to $\mathbb Z/3\mathbb Z$ is a <group homomorphism>: reducing a <sum> agrees with adding the reduced residues. For example, $2+5=1$ modulo six, and both sides reduce to one modulo three; the outputs $2$ and $2$ also add to one modulo three. This <map> is <surjective> but not <injective>, since zero and three both map to zero.
The <sign of a permutation> gives another example, from the <symmetric group> $S_3$ to the multiplicative <group> $\{1,-1\}$. A product of two odd <permutations> is even, matching $(-1)(-1)=1$. Its <group kernel> is $A_3$, the <alternating group> of even <permutations>. This shows how a <group homomorphism> can retain a useful feature while forgetting other details.
Back to article page
