Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 9 / i

A subgroup $N\le G$ is a normal subgroup when conjugation by every group element preserves it:Equivalently, $gN=Ng$ for every $g$; normality permits the cosets to inherit a quotient-group multiplication.

A group homomorphism is a map preserving multiplication:It consequently sends identity to identity and inverses to inverses, since $\varphi (e)=\varphi (e{)}^2$ and $\varphi (g)\varphi (g^{-1})=\varphi (e)$.