# 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 ${\displaystyle G}$ is called powerful if the commutator subgroup ${\displaystyle [G,G]}$ is contained in the subgroup ${\displaystyle G^{p}=\langle g^{p}|g\in G\rangle }$ for odd ${\displaystyle p}$, or if ${\displaystyle [G,G]}$ is contained in the subgroup ${\displaystyle G^{4}}$ 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 ${\displaystyle G}$ is a powerful p-group then:

Some less abelian-like properties are: if ${\displaystyle G}$ is a powerful p-group then:

## References

• Lazard, Michel (1965), Groupes analytiques p-adiques, Publ.Math.IHES 26 (1965), 389-603.
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