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).

Formal definition[edit]

A finite p-group G is called powerful if the commutator subgroup [G,G] is contained in the subgroup G^p = \langle g^p | g\in G\rangle for odd p, or if [G,G] is contained in the subgroup G^4 for p=2.

Properties of powerful p-groups[edit]

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 \Phi(G) of G has the property \Phi(G) = G^p.
  • G^{p^k} = \{g^{p^k}|g\in G\} for all k\geq 1. That is, the group generated by pth powers is precisely the set of pth powers.
  • If G = \langle g_1, \ldots, g_d\rangle then G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle for all k\geq 1.
  • The kth entry of the lower central series of G has the property \gamma_k(G)\leq G^{p^{k-1}} for all k\geq 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.