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 pgroups. One of the first standard results using the class equation is that the center of a finite pgroup cannot be the trivial subgroup. More generally, every finite pgroup is both nilpotent and solvable.
pgroups of the same order are not necessarily isomorphic; for example, the cyclic group C_{4} and the Klein group V_{4} are both 2groups of order 4, but they are not isomorphic. Nor need a pgroup be abelian; the dihedral group D_{8} is a nonabelian 2group.
In an asymptotic sense, almost all finite groups are pgroups. In fact, almost all finite groups are 2groups. 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 2group tends to 1 as n tends to infinity. For instance, if n = 2000, then the probability of picking a 2group of order 1024 is greater than 99%.
Every nontrivial finite group contains a subgroup which is a pgroup. 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 pgroup, 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.
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