After defining p-groups and p-subgroups, we now prove the first structural theorem about finite p-groups. The focus keyword is centers of finite p-groups, because the center is the place where a nonabelian group still behaves commutatively. Every nontrivial finite p-group has a nontrivial center, and this fact drives many later conclusions about groups of order $p^2$, groups of order $p^3$, normal subgroups, and Sylow arguments.

Let $G$ be a group. The \textbf{center} of $G$ is the set $$\begin{equation*} Z(G)=\{z\in G:zg=gz\text{ for all }g\in G\}. \end{equation*}$$ Elements of $Z(G)$ commute with every element of $G$.