Misplaced Pages

Powerful p-group

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Powerful p-groups)

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 } 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 } 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

Categories: