Powerful p-group

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 formula_1 is called powerful if the commutator subgroup formula_2 is contained in the subgroup formula_3 for odd formula_4, or if formula_2 is contained in the subgroup formula_6 for formula_7.
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 formula_1 is a powerful "p"-group then: 
Some less abelian-like properties are: if formula_1 is a powerful "p"-group then: 