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

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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  .

Properties of powerful p-groups

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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   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.

References

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  • 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