Powerful p-group
= Powerful p-group
{wiki}
A <finite p-group> is powerful if $[G,G]\leq G^p$ for odd $p$, or $[G,G]\leq G^4$ for $p=2$. Here powers denote generated <subgroups>. The same definition applies to <pro-p groups> using closed power and <commutator subgroups>. Its <Frattini subgroup> is $G^p$. For odd $p$, successive power layers admit linear correction of roots, which makes every element of $G^p$ an actual $p$th power.