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

Formal definition

A finite p-group $G$ is called powerful if the commutator subgroup $[G,G]$ is contained in the subgroup $G^p=⟨g^p|g\in G⟩$ 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: 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 $\mathrm{\Phi }(G)$ of $G$ has the property $\mathrm{\Phi }(G)=G^p.$
• $G^{p^k}=\{g^{p^k}|g\in G\}$ for all $k\ge 1.$ That is, the group generated by $p$th powers is precisely the set of $p$th powers.
• If $G=⟨g_1,\dots ,g_d⟩$ then $G^{p^k}=⟨g_1^{p^k},\dots ,g_d^{p^k}⟩$ for all $k\ge 1.$
• The $k$th entry of the lower central series of $G$ has the property ${\gamma }_k(G)\le G^{p^{k-1}}$ for all $k\ge 1.$
• 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

• Lazard, Michel (1965), Groupes analytiques p-adiques, Publ. Math. IHÉS 26 (1965), 389–603.
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