P-group

For a prime number $p$, a finite p-group is a group of order $p^n$ for some nonnegative integer $n$. More generally, a possibly infinite group is called a p-group when every element has finite order a power of $p$; finiteness should therefore be stated when group order is used.

Examples are the cyclic group of order $p^2$ and the dihedral group of order eight for $p=2$. Every nontrivial finite p-group has a nontrivial center of a group: its class equation shows that noncentral conjugacy classes have sizes divisible by $p$, forcing the size of the center of a group to be divisible by $p$. This makes induction on the order particularly effective.