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 A finite p-group G {\displaystyle G} is called powerful if the commutator subgroup [ G , G ] {\displaystyle [G,G]} is contained in the subgroup G p = ⟨ g p | g ∈ G ⟩ {\displaystyle G^{p}=\langle g^{p}|g\in G\rangle } for odd p {\displaystyle p} , or if [ G , G ] {\displaystyle [G,G]} is contained in the subgroup G 4 {\displaystyle G^{4}} for p = 2 {\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 G {\displaystyle G} is a powerful p-group then:
The Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of G {\displaystyle G} has the property Φ ( G ) = G p . {\displaystyle \Phi (G)=G^{p}.}
G p k = { g p k | g ∈ G } {\displaystyle G^{p^{k}}=\{g^{p^{k}}|g\in G\}} for all k ≥ 1. {\displaystyle k\geq 1.} That is, the group generated by p {\displaystyle p} th powers is precisely the set of p {\displaystyle p} th powers. If G = ⟨ g 1 , … , g d ⟩ {\displaystyle G=\langle g_{1},\ldots ,g_{d}\rangle } then G p k = ⟨ g 1 p k , … , g d p k ⟩ {\displaystyle G^{p^{k}}=\langle g_{1}^{p^{k}},\ldots ,g_{d}^{p^{k}}\rangle } for all k ≥ 1. {\displaystyle k\geq 1.}
The k {\displaystyle k} th entry of the lower central series of G {\displaystyle G} has the property γ k ( G ) ≤ G p k − 1 {\displaystyle \gamma _{k}(G)\leq G^{p^{k-1}}} for all k ≥ 1. {\displaystyle k\geq 1.}
Every quotient group of a powerful p-group is powerful. The Prüfer rank of G {\displaystyle G} is equal to the minimal number of generators of G . {\displaystyle G.}
Some less abelian-like properties are: if G {\displaystyle G} is a powerful p-group then:
G p k {\displaystyle G^{p^{k}}} is powerful. Subgroups of G {\displaystyle G} are not necessarily powerful.
References Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHÉS 26 (1965), 389–603. Dixon, J. D.; du Sautoy, M. P. F.; Mann, A.; Segal, D. (1991), Analytic pro-p-groups, Cambridge University Press, ISBN 0-521-39580-1, MR 1152800 Khukhro, E. I. (1998), p-automorphisms of finite p-groups, Cambridge University Press, doi:10.1017/CBO9780511526008, ISBN 0-521-59717-X, MR 1615819 Leedham-Green, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press, ISBN 978-0-19-853548-5, MR 1918951 Lubotzky, Alexander; Mann, Avinoam (1987), "Powerful p-groups. I. Finite Groups", J. Algebra, 105 (2): 484–505, doi:10.1016/0021-8693(87)90211-0, MR 0873681 Vaughan-Lee, Michael (1993), The restricted Burnside problem (2nd ed.), Oxford University Press, ISBN 0-19-853786-7, MR 1364414