Let G {\displaystyle {}G} and H {\displaystyle {}H} denote groups, and let φ : G → H {\displaystyle {}\varphi \colon G\rightarrow H} be a group homomorphism.
φ ( e G ) = e H {\displaystyle {}\varphi (e_{G})=e_{H}} and ( φ ( g ) ) − 1 = φ ( g − 1 ) {\displaystyle {}(\varphi (g))^{-1}=\varphi {\left(g^{-1}\right)}} for every
and 
denote
groups,
and let
be a
group homomorphism.
and
for every
