Having proved Cauchy's theorem and studied subgroup existence in finite abelian groups, we now isolate groups whose order is governed by a single prime. The focus keyword is p-groups and p-subgroups, because this language is the gateway to Sylow theory. A p-group is built entirely from powers of one prime, and a p-subgroup is the same idea inside a larger group. Students often confuse an element of p-power order with a p-subgroup; this lesson separates the two ideas carefully and shows why both are essential.

Let $p$ be a prime number and let $G$ be a finite group. Then $G$ is called a \textbf{p-group} if there exists an integer $n\geq0$ such that $$\begin{equation*} |G|=p^n. \end{equation*}$$ The group with one element is a p-group for every prime $p$, corresponding to $n=0$.