Definition 9

A finite group whose order is a power of $p$ is called a $p$-group.

Let $G$ be a $p$-group of order $p^r$. Every divisor of $p^r$ is a power of $p$ ($§ 4$, no. 10, Corollary to Theorem 7). Therefore every subgroup and every quotient group of $G$ is a $p$-group ($§ 4$, no. 4, Corollary to Proposition 4); the cardinal of every homogeneous space of $G$ is a power of $p$ ($§ 5$, no. 5, Theorem 1).

An extension of a $p$-group by a $p$-group is a $p$-group.

Examples. (1) A commutative $p$-group is isomorphic to a product of cyclic groups $\mathbf{Z}/p^n\mathbf{Z}$ (cf. Exercise 19 and also VII, $§ 4$, no. 7, Proposition 7). (2) Let $k$ be a finite field of characteristic $p$. The strict triangular group $T_1(n, k)$ is a $p$-group. (3) The quaternionic group $\{ \pm 1, \pm i, \pm j, \pm k \}$ is a 2-group (cf. Exercise 4).