Groups in which every Lagrange subset is a factor

Abstract

We determine the finite groups $G$ in which every subset $A\subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B\subseteq G$ of cardinality $|G|/|A|$ such that $G=A\cdot B$ or $G=B\cdot A$.