p-residual of a finite group ($O^p(G)$)

The p-residual is the smallest normal subgroup with p-group quotient, equivalently the subgroup generated by all elements of order prime to $p$. It is characteristic and satisfies $O^p(O^p(G))=O^p(G)$: an extension of two finite p-groups is a p-group. A finite group has a normal p-complement exactly when its p-residual has order prime to $p$.