In mathematics, given a prime number p, a pgroup is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power p^{n} is equal to the identity element. Such groups are also called pprimary or simply primary.
A finite group is a pgroup if and only if its order (the number of its elements) is a power of p. The remainder of this article deals with finite pgroups. For an example of an infinite abelian pgroup, see Prüfer group, and for an example of an infinite simple pgroup, see Tarski monster group.
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Properties
Quite a lot is known about the structure of finite pgroups.
Nontrivial center
One of the first standard results using the class equation is that the center of a nontrivial finite pgroup cannot be the trivial subgroup (proof).
This forms the basis for many inductive methods in pgroups.
For instance, the normalizer N of a proper subgroup H of a finite pgroup 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 pgroup is nilpotent.
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