In mathematics, in the field of group theory, especially in the study of p-groups and pro-p-groups, the concept of powerful p-groups plays an important role. They were introduced in, where a number of applications are given, including results on Schur multipliers. Powerful p-groups are used in the study of automorphisms of p-groups, the solution of the restricted Burnside problem , the classification of finite p-groups via the coclass conjectures , and provided an excellent method of understanding analytic pro-p-groups.

Formal definition

A finite p-group G is called powerful if the commutator subgroup [G,G] is contained in the subgroup for odd p, or if [G,G] is contained in the subgroup G^4 for p=2.

Properties of powerful p-groups

Powerful p-groups have many properties similar to abelian groups, and thus provide a good basis for studying p-groups. Every finite p-group can be expressed as a section of a powerful p-group. Powerful p-groups are also useful in the study of pro-p groups as it provides a simple means for characterising p-adic analytic groups (groups that are manifolds over the p-adic numbers): A finitely generated pro-p group is p-adic analytic if and only if it contains an open normal subgroup that is powerful: this is a special case of a deep result of Michel Lazard (1965). Some properties similar to abelian p-groups are: if G is a powerful p-group then: Some less abelian-like properties are: if G is a powerful p-group then: