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, 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:
 * 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 $$p$$th powers is precisely the set of $$p$$th powers.
 * If $$G = \langle g_1, \ldots, g_d\rangle$$ then $$G^{p^k} = \langle g_1^{p^k},\ldots,g_d^{p^k}\rangle$$ 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.