After studying homomorphisms from cyclic groups, it is natural to examine functions that send each element of a group to one of its powers. The focus keyword power maps on groups refers to maps such as $\phi(a)=a^n$. These maps are easy to write but not always homomorphisms. In a commutative group, powers of a product split as expected, so the map $a\mapsto a^n$ preserves multiplication. In a noncommutative group, the expression $(ab)^n$ may not equal $a^nb^n$. This lesson separates the reliable abelian case from the dangerous non-abelian case and proves a useful isomorphism criterion for finite commutative groups.

Let $G$ be a group and let $n$ be an integer. The \textbf{power map} with exponent $n$ is the function $\phi_n:G\to G$ defined by $$\begin{equation*} \phi_n(a)=a^n \end{equation*}$$ for all $a\in G$, whenever the exponent notation is interpreted in the group $G$.