P-group

In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pⁿ copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.

Abelian p-groups are also called p-primary or simply primary.

A finite group is a p-group if and only if its order (the number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee the existence of a subgroup of G of order pⁿ for every prime power pⁿ that divides the order of G.

Every finite p-group is nilpotent.

The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.

Every p-group is periodic since by definition every element has finite order.

If p is prime and G is a group of order p^k, then G has a normal subgroup of order p^m for every 1 ≤ m ≤ k. This follows by induction, using Cauchy's theorem and the Correspondence Theorem for groups. A proof sketch is as follows: because the center Z of G is non-trivial (see below), according to Cauchy's theorem Z has a subgroup H of order p. Being central in G, H is necessarily normal in G. We may now apply the inductive hypothesis to G/H, and the result follows from the Correspondence Theorem.

One of the first standard results using the class equation is that the center of a non-trivial finite p-group cannot be the trivial subgroup.