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.

Not to be confused with n-group (category theory).
Algebraic structure → Group theory
Group theory
Basic notions
Group homomorphisms
Finite groups
Classification of finite simple groups
Modular groups
  • PSL(2, )
  • SL(2, )
Topological and Lie groups
Infinite dimensional Lie group
  • O(∞)
  • SU(∞)
  • Sp(∞)
Algebraic groups

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.

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