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 is called powerful if the commutator subgroup is contained in the subgroup for odd , or if is contained in the subgroup for . 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 is a powerful p-group then: The Frattini subgroup of has the property for all That is, the group generated by th powers is precisely the set of th powers. If then for all The th entry of the lower central series of has the property for all Every quotient group of a powerful p-group is powerful. The Prüfer rank of is equal to the minimal number of generators of Some less abelian-like properties are: if is a powerful p-group then: is powerful. Subgroups of are not necessarily powerful. References Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965), 389–603. P-groups Properties of groups