A simultaneous variant of the Hall-Paige theorem and applications

For $k\ge 1$ , we say that a finite group $G$ satisfies property ${\mathcal{P}}_k$ if, for every ${\beta }_1,\dots ,{\beta }_k\in \mathrm{Aut}(G)$, there exists a bijection $f$ of $G$ such that each map $g\mapsto {\beta }_i(g)f(g)$ is a bijection. For $k=1$ this is equivalent to the classical Hall–Paige complete mapping property. We prove that if $p$ is the smallest prime divisor of $\mathrm{\mid }G\mathrm{\mid }$, then $G$ satisfies ${\mathcal{P}}_k$ for every $k<p-1$ and we show that this bound is sharp. For finite abelian $p$-groups, the bijection $f$ may moreover be chosen to be an automorphism, and for noncyclic homocyclic $p$-groups this remains true for $k=p-1$. Finally, we interpret ${\mathcal{P}}_k$ in terms of common transversals and common orthogonal mates of Latin squares generated from automorphisms and the Cayley table of a finite group $G$.