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 , 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 , the solution of the restricted Burnside problem , the classification of finite ''p''-groups via the coclass conjectures , and provided an excellent method of understanding analytic pro-''p''-groups .

Formal definition

A finite ''p''-group $G$ is called powerful if the commutator subgroup ,G/math> is contained in the subgroup $G^p = \langle g^p , g\in G\rangle$ for odd $p$, or if ,G/math> 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: this is a special case of a deep result of Michel Lazard (1965).

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^ = \$ 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 $G^ = \langle g_1^,\ldots,g_d^\rangle$ for all $k\geq 1.$
* The $k$th entry of the lower central series of $G$ has the property $\gamma_k(G)\leq G^$ 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:
* $G^$ is powerful.
* Subgroups of $G$ are not necessarily powerful.

References

* Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965), 389–603.
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* {{citation , mr=1364414 , last1=Vaughan-Lee , first1=Michael , title=The restricted Burnside problem , edition=2nd , publisher=Oxford University Press , year=1993 , isbn=0-19-853786-7

P-groups
Properties of groups