Harman Patil (Editor)

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 1987), 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 2002), and provided an excellent method of understanding analytic pro-p-groups (Dixon et al. 1991).

Contents

Formal definition

A finite p-group G is called powerful if the commutator subgroup [ G , G ] is contained in the subgroup G p = ⟨ g p | g ∈ G ⟩ for odd p , or if [ G , G ] is contained in the subgroup 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 G is a powerful p-group then:

  • The Frattini subgroup Φ ( G ) of G has the property Φ ( G ) = G p .
  • G p k = { g p k | g ∈ G } for all k ≥ 1. That is, the group generated by p th powers is precisely the set of p th powers.
  • If G = ⟨ g 1 , … , g d ⟩ then G p k = ⟨ g 1 p k , … , g d p k ⟩ for all k ≥ 1.
  • The k th entry of the lower central series of G has the property γ k ( G ) ≤ G p k − 1 for all k ≥ 1.
  • Every quotient group of a powerful p-group is powerful.
  • The Prüfer rank of G is equal to the minimal number of generators of G .

    • Some less abelian-like properties are: if G is a powerful p-group then:

  • G p k is powerful.
  • Subgroups of G are not necessarily powerful.

    • References

    Powerful p-group Wikipedia
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