p-group

In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, 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. In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, 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, for every prime power pn that divides the order of G, the existence of a subgroup of G of order pn. 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 pk, then G has a normal subgroup of order pm 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 nontrivial (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. This forms the basis for many inductive methods in p-groups. For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H=N, the center Z is contained in N, and so also in H, but then there is a smaller example H/Z whose normalizer in G/Z is N/Z=H/Z, creating an infinite descent. As a corollary, every finite p-group is nilpotent.