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 harv|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 harv|Khukro|1998, the solution of the restricted Burnside problem harv|Vaughan-Lee|1993, the classification of finite p-groups via the coclass conjectures harv|Leedham-Green|McKay|2002, and provided an excellent method of understanding analytic pro-p-groups harv|Dixon|du Sautoy|Mann|Segal|1991.

Formal definition

A finite p-group $G$ is called powerful if the commutator subgroup $[G,G]$ is contained in the subgroup $G^p\; =\; langle\; g^p\; |\; gin\; G\; angle$ for odd $p$, or if $[G,G]$ is contained in the subgroup $G^4$ for "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.

Some properties similar to abelian "p"-groups are: if $G$ is a powerful "p"-group then:
* The Frattini subgroup $Phi(G)$ of $G$ has the property $Phi(G)\; =\; G^p.$
* $G^\{p^k\}\; =\; \{g^\{p^k\}|gin\; G\}$ for all $kgeq\; 1.$ That is, the "group generated" by $p$th powers is precisely the "set" of $p$th powers.
* If $G\; =\; langle\; g\_1,\; ldots,\; g\_d\; angle$ then $G^\{p^k\}\; =\; langle\; g\_1^\{p^k\},ldots,g\_d^\{p^k\}\; angle$ for all $kgeq\; 1.$
* The $k$th entry of the lower central series of $G$ has the property $gamma\_k(G)leq\; G^\{p^\{k-1$ for all $kgeq\; 1.$
* 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.

References

* | last1=Dixon | first1=J. D. | last2=du Sautoy | first2=M. P. F. | author2-link=Marcus duSautoy | last3=Mann | first3=A. | last4=Segal | first4=D. | title=Analytic pro-p-groups | publisher=Cambridge University Press | year=1991 | ISBN=0-521-39580-1 | author2-link=Marcus du Sautoy
* | last1=Khukhro | first1=E. I. | title=p-automorphisms of finite p-groups | publisher=Cambridge University Press | year=1998 | ISBN=0-521-59717-X
*Citation | last1=Leedham-Green | first1=C. R. | author1-link=Charles Leedham-Green | last2=McKay | first2=Susan | title=The structure of groups of prime power order | publisher=Oxford University Press | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853548-5 | id=MathSciNet | id = 1918951 | year=2002 | volume=27
* | last1=Lubotzky | first1=Alexander | last2=Mann | first2=Avinoam | title=Powerful p-groups. I. Finite Groups | journal=J. Algebra | volume=105 | year=1987 | pages=484–505
* | last1=Vaughan-Lee | first1=Michael | title=The restricted Burnside problem. | edition=2nd | publisher=Oxford University Press | year=1993 | ISBN=0-19-853786-7