P-group

Definition

Let $p$ be a prime number. A group $G$ is a $p$-group when, for each $g \in G$, the order of $g$ is $p^n$ for some integer $n \geq 0$.

Remarks

A finite group is a $p$-group if and only if $|G| = p^k$ for some $k \geq 0$. One direction uses Lagrange's theorem, and the other direction uses Cauchy's theorem. The dihedral group $D_4$ of order $8$ is a $2$-group that is not abelian. The trivial group is a $p$-group for each prime $p$.

Explanations

L1 human

In a $p$-group, each element comes back to the identity after a power of $p$ steps. For finite groups this means that the number of elements is a power of $p$. These groups are the building blocks of the Sylow theory.