Exercise 2.1.12 (Kernel and image of $a \mapsto a^n$)

Proof. Let $n\in$ be some fixed integer. Consider the map

Since $A$ is abelian, for all $x,y\in A$,

so $\phi$ is a homomorphism.

(a)

Put H = $\{a^n\mid a\in A\}$. Then for all $b\in A$, $$b\in H⟺\exists <!--\; FUNCTION\; APPLICATION\; -->a\in A,b=a^n⟺g\in \phi (A),$$

so $H=\phi (A)$ is a subgroup of $A$.

(b)

Put $K=\{a\in A\mid a^n=1\}$. Then for all $a\in A$, $$a\in K⟺a^n=1⟺a\in \ker <!--\; FUNCTION\; APPLICATION\; -->(\phi ),$$

so $K=\ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )$ is a subgroup of $A$.