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 (Lubotzky & Mann 1987), 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 (Khukhro 1998), the solution of the restricted Burnside problem (Vaughan-Lee 1993), the classification of finite p-groups via the coclass conjectures (Leedham-Green & McKay 2002), and provided an excellent method of understanding analytic pro-p-groups (Dixon et al. 1991).
is called powerful if the commutator subgroup
Powerful p-groups have many properties similar to abelian groups, and thus provide a good basis for studying p-groups.
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
is a powerful p-group then: Some less abelian-like properties are: if