Powerful p-group

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.

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^{p^k} = {g^{p^k}|g\in G} for all k\geq 1. That is, the group generated by pth powers is precisely the set of pth 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 kth 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.

References

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