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 (Khukro 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).

[edit] Formal definition

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.

[edit] 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.

Some properties similar to abelian p-groups are: if $G$ is a powerful p-group then:

The Frattini subgroup $Φ(G)$ of $G$ has the property $Φ(G) = G p .$
$G^{p^k} = \{g^{p^k}|g\in G\}$ for all $k\geq 1.$ That is, the group generated by $p$ th powers is precisely the set of $p$ th 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 $k$ th 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.

[edit] References

Dixon, J. D.; du Sautoy, M. P. F.; Mann, A. & Segal, D. (1991), Analytic pro-p-groups, Cambridge University Press, MR 1152800, ISBN 0-521-39580-1
Khukhro, E. I. (1998), p-automorphisms of finite p-groups, Cambridge University Press, MR 1615819, ISBN 0-521-59717-X
Leedham-Green, C. R. & McKay, S. (2002), The structure of groups of prime power order, Oxford University Press, MR 1918951, ISBN 0-19-853548-1
Lubotzky, Alexander & Mann, Avinoam (1987), “Powerful p-groups. I. Finite Groups”, J. Algebra 105: 484–505, MR 0873681
Vaughan-Lee, Michael (1993), The restricted Burnside problem. (2nd ed.), Oxford University Press, MR 1364414, ISBN 0-19-853786-7