Powerful p-group

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 ${\displaystyle G}$ is called powerful if the commutator subgroup ${\displaystyle [G,G]}$ is contained in the subgroup ${\displaystyle G^{p}=\langle g^{p}|g\in G\rangle }$ for odd ${\displaystyle p}$, or if ${\displaystyle [G,G]}$ is contained in the subgroup ${\displaystyle G^{4}}$ for ${\displaystyle p=2}$.

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 ${\displaystyle G}$ is a powerful p-group then:

• The Frattini subgroup ${\displaystyle \Phi (G)}$ of ${\displaystyle G}$ has the property ${\displaystyle \Phi (G)=G^{p}.}$
• ${\displaystyle G^{p^{k}}=\{g^{p^{k}}|g\in G\}}$ for all ${\displaystyle k\geq 1.}$ That is, the group generated by ${\displaystyle p}$th powers is precisely the set of ${\displaystyle p}$th powers.
• If ${\displaystyle G=\langle g_{1},\ldots ,g_{d}\rangle }$ then ${\displaystyle G^{p^{k}}=\langle g_{1}^{p^{k}},\ldots ,g_{d}^{p^{k}}\rangle }$ for all ${\displaystyle k\geq 1.}$
• The ${\displaystyle k}$th entry of the lower central series of ${\displaystyle G}$ has the property ${\displaystyle \gamma _{k}(G)\leq G^{p^{k-1}}}$ for all ${\displaystyle k\geq 1.}$
• Every quotient group of a powerful p-group is powerful.
• The Prüfer rank of ${\displaystyle G}$ is equal to the minimal number of generators of ${\displaystyle G.}$

Some less abelian-like properties are: if ${\displaystyle G}$ is a powerful p-group then:

• ${\displaystyle G^{p^{k}}}$ is powerful.
• Subgroups of ${\displaystyle G}$ are not necessarily powerful.

References

• 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, 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, 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