# 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 Template:Harv, 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 Template:Harv, the solution of the restricted Burnside problem Template:Harv, the classification of finite

*p*-groups via the coclass conjectures Template:Harv, and provided an excellent method of understanding analytic pro-

*p*-groups Template:Harv.

## 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 *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 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.
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