# 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 (Lubotzky Mann), 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), and provided an excellent method of understanding analytic pro-p-groups (Dixon du Sautoy).

## 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 $\displaystyle{ p=2 }$.

## 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:

• The Frattini subgroup $\displaystyle{ \Phi(G) }$ of $\displaystyle{ G }$ has the property $\displaystyle{ \Phi(G) = G^p. }$
• $\displaystyle{ G^{p^k} = \{g^{p^k}|g\in G\} }$ for all $\displaystyle{ k\geq 1. }$ That is, the group generated by $\displaystyle{ p }$th powers is precisely the set of $\displaystyle{ p }$th powers.
• If $\displaystyle{ G = \langle g_1, \ldots, g_d\rangle }$ then $\displaystyle{ G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle }$ for all $\displaystyle{ k\geq 1. }$
• The $\displaystyle{ k }$th entry of the lower central series of $\displaystyle{ G }$ has the property $\displaystyle{ \gamma_k(G)\leq G^{p^{k-1}} }$ for all $\displaystyle{ k\geq 1. }$
• Every quotient group of a powerful p-group is powerful.
• The Prüfer rank of $\displaystyle{ G }$ is equal to the minimal number of generators of $\displaystyle{ G. }$

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

• $\displaystyle{ G^{p^k} }$ is powerful.
• Subgroups of $\displaystyle{ G }$ are not necessarily powerful.