If *G* is finite, this is equivalent to requiring that the order of *G* (the number of its elements) itself be a power of *p*. Quite a lot is known about the structure of finite *p*-groups. One of the first standard results using the class equation is that the center of a finite *p*-group cannot be the trivial subgroup. More generally, every finite *p*-group is both nilpotent and solvable.

*p*-groups of the same order are not necessarily isomorphic; for example, the cyclic group *C*_{4} and the Klein group *V*_{4} are both 2-groups of order 4, but they are not isomorphic. Nor need a *p*-group be abelian; the dihedral group *D*_{8} is a non-abelian 2-group.

In an asymptotic sense, almost all finite groups are *p*-groups. In
fact, almost all finite groups are 2-groups. The sense taken is
that if you fix a number *n* and choose uniformly randomly from
a list of all the isomorphism classes
of groups of order at most *n*,
then the probability that you pick a 2-group tends to 1 as *n*
tends to infinity. For instance, if *n* = 2000, then the probability
of picking a 2-group of order 1024 is greater than 99%.

Every non-trivial finite group contains a subgroup which is a *p*-group. The details are described the Sylow theorems.

For an infinite example, let *G* be the set of rational numbers of the form \*m*/*p*^{n} where *m* and *n* are natural numbers and *m* < *p*^{n}. This set becomes a group if we perform addition modulo 1. *G* is an infinite abelian *p*-group, and any group isomorphic to *G* is called a ** p^{∞}-group**. Groups of this type are important in the classification of infinite abelian groups.

The *p*^{∞}-group can alternatively be described as the multiplicative subgroup of **C** - {0} consisting of all *p*^{n}-th roots of unity, or as the direct limit of the groups **Z** / *p*^{n}**Z** with respect to the homomorphisms **Z** / *p*^{n}**Z** → **Z** / *p*^{n+1}**Z** which are induced by multiplication with *p*.