Powerful p-group explained

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,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: \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 p th powers is precisely the set of p th powers.

• If

G = \langle g _{1,} \ldots, g _{d\rangle} then G ^{p ^{k}} = \langle g ^{p ^{k}} _{1} ,\ldots,g ^{p ^{k}} _{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 ^{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.