The nontrivial center theorem becomes especially powerful when the order of the group is small. The focus keyword is groups of order p squared and p cubed, because these are the first cases where p-group structure can be described with strong conclusions. Groups of order $p^2$ are always abelian, while groups of order $p^3$ may be abelian or nonabelian but still have a large center. These examples show how counting, centers, and quotient groups cooperate in finite group theory.

Let $p$ be a prime number. A finite abelian p-group $G$ is called \textbf{elementary abelian} if every nonidentity element of $G$ has order $p$. In this case $G$ behaves like a vector space over the field $\mathbb{Z}_p$.