Lower p-series ($P_1(G)=G,P_{i+1}(G)=P_i(G{)}^p[P_i(G),G]$)

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.