Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 8 / a / Solution

Let $n=\mathrm{ord}(a)$, which is finite. A group homomorphism preserves powers and the identity, soIf $r=\mathrm{ord}(f(a))$, divide $n$ by $r$: $n=qr+s$ with $0\le s<r$. Then $e^{\prime }=f(a{)}^n=f(a{)}^s$, and minimality of $r$ forces $s=0$. ThusIf $f$ is a group isomorphism, apply the same argument to its inverse: $\mathrm{ord}(a)\le \mathrm{ord}(f(a))$. The two inequalities yield