Powerful p-group

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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.

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.

[edit] Properties of Powerful p-groups

Some properties similar to Abelian p-groups are: if $G$ is a powerful p-group then:

The Frattini subgroup $Φ(G)$ of $G$ has the property $Φ(G) = G p .$
$G^{p^k} = \{g^{p^k}|g\in G\}$ 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 $G .$