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).
A finite p-group is called powerful if the commutator subgroup is contained in the subgroup for odd , or if is contained in the subgroup for p=2.
Properties of powerful p-groups
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 is a powerful p-group then:
- The Frattini subgroup of has the property
- for all That is, the group generated by th powers is precisely the set of th powers.
- If then for all
- The th entry of the lower central series of has the property for all
- Every quotient group of a powerful p-group is powerful.
- The Prüfer rank of is equal to the minimal number of generators of
Some less abelian-like properties are: if is a powerful p-group then:
- is powerful.
- Subgroups of are not necessarily powerful.
- Lazard, Michel (1965), Groupes analytiques p-adiques, Publ.Math.IHES 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, MR1152800
- Khukhro, E. I. (1998), p-automorphisms of finite p-groups, Cambridge University Press, ISBN 0-521-59717-X, MR1615819
- Leedham-Green, C. R.; McKay, Susan (2002), The structure of groups of prime power order, London Mathematical Society Monographs. New Series 27, Oxford University Press, ISBN 978-0-19-853548-5, MR1918951
- 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, MR0873681
- Vaughan-Lee, Michael (1993), The restricted Burnside problem (2nd ed.), Oxford University Press, ISBN 0-19-853786-7, MR1364414