# P-group

In mathematics, given a prime number *p*, a ** p-group** is a periodic group in which each element has a power of

*p*as its order. That is, for each element

*g*of the group, there exists a natural number

*n*such that

*g*to the power

*p*is equal to the identity element. Such groups are also called

^{n}**primary**.

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 non-trivial finite *p*-group cannot be the trivial subgroup. More generally, every finite *p*-group is nilpotent, and therefore 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 *Dih*_{4} of order 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 fraction of isomorphism classes of 2-groups
among isomorphism classes of groups of order at most *n* tends to 1 as *n* tends to infinity. For instance, more than 99% of all different groups of order at most 2000 are 2-groups of order 1024.

Every non-trivial finite group contains a subgroup which is a *p*-group. The details are described in 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** (or

**quasicyclic**, or

*p*-group**Prüfer**). Groups of this type are important in the classification of infinite abelian groups.

*p*-groupThe *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*.