p-residual of a finite group
= p-residual of a finite group
{title2=$O^p(G)$}
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 $p$.