Powerful p-group

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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 [math]\displaystyle{ G }[/math] is called powerful if the commutator subgroup [math]\displaystyle{ [G,G] }[/math] is contained in the subgroup [math]\displaystyle{ G^p = \langle g^p | g\in G\rangle }[/math] for odd [math]\displaystyle{ p }[/math], or if [math]\displaystyle{ [G,G] }[/math] is contained in the subgroup [math]\displaystyle{ G^4 }[/math] for [math]\displaystyle{ p=2 }[/math].

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 [math]\displaystyle{ G }[/math] is a powerful p-group then:

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

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

  • [math]\displaystyle{ G^{p^k} }[/math] is powerful.
  • Subgroups of [math]\displaystyle{ G }[/math] are not necessarily powerful.