Hall–Paige conjecture on transversals of Cayley tables

For a finite group $G$, let $L(G)$ be its Cayley table. The Hall–Paige condition is the condition that the sum of the elements of $G$ is the identity in the abelianisation $G^{\mathrm{ab}}=G/G'$, equivalently that every Sylow $2$-subgroup of $G$ is trivial or non-cyclic. A transversal of $L(G)$ selects one entry from each row, column, and group element. Hall–Paige conjecture. For any finite group $G$, $L(G)$ has a transversal if and only if $G$ satisfies the Hall–Paige condition. The conjecture was proved in 2009 using the classification of finite simple groups, in work of Wilcox, Evans and Bray.