Powerful P-group - Properties of Powerful p-groups

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 is a powerful p-group then:

• The Frattini subgroup of has the property
• for all That is, the group generated by th powers is precisely the set of th powers.
• If then for all
• The th entry of the lower central series of has the property for all
• Every quotient group of a powerful p-group is powerful.
• The Prüfer rank of is equal to the minimal number of generators of

Some less abelian-like properties are: if is a powerful p-group then:

• is powerful.
• Subgroups of are not necessarily powerful.