P-group

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The correct title of this article is p-group. The initial letter is shown capitalized due to technical restrictions.

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 nonnegative integer n such that g to the power pⁿ is equal to the identity element. Such groups are also called 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. A finite p-group with order pⁿ contains subgroups of order pⁱ with 0 ≤ i ≤ n. 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 C₄ and the Klein group V₄ are both 2-groups of order 4, but they are not isomorphic. Nor need a p-group be abelian; the dihedral group Dih₄ of order 8 is a non-abelian 2-group. (However, every group of order p² is abelian.)

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. ^[1]

Every non-trivial finite group contains a subgroup which is a non-trivial p-group. The details are described in the Sylow theorems.

For an infinite example, see Prüfer group.