p-group

Fix a prime number $p$. A $p$-group is a group $G$ such that every element $g\in G$ has order $p^k$ for some integer $k\ge 0$ (depending on $g$). If $G$ is finite, this is equivalent to saying that the cardinality of $G$ is a power of $p$, i.e. $|G|=p^n$ for some $n\ge 0$.

Finite $p$-groups have strong structure properties; for instance, they have nontrivial center (see p-group has nontrivial center ). Maximal $p$-subgroups of a finite group occur as Sylow p-subgroups , central in Sylow theory.

Examples:

• The additive group $\mathbb{Z}/p^n\mathbb{Z}$ is a finite $p$-group of order $p^n$.
• The quaternion group $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$ is a $2$-group (every element has order $1$, $2$, or $4$).
• The trivial group $\{e\}$ is a $p$-group for every prime $p$ (it has order $p^0=1$).