Lower p-series
= Lower p-series
{title2=$P_1(G)=G,\quad P_{i+1}(G)=P_i(G)^p[P_i(G),G]$}
The displayed descending <filtration> combines powers and <group commutators>. In a <pro-p group>, take the <closure> of the generated product at each step. For a <powerful p-group> these layers are $P_i(G)=G^{p^{i-1}}$, with iterated closed power <subgroups> in the infinite case. The successive quotients are <elementary abelian p-groups>.