Verifying a conjecture of L. Rédei for $p=13$

Abstract:

In 1970 L. Rédei conjectured that if an elementary $p$-group $G$ of order $p^3$ is a direct product of its subsets $A$ and $B$ such that both $A$ and $B$ contain the identity element of $G$, then at least one of the factors $A$ and $B$ cannot span the whole $G$. We will verify this conjecture for $p=13$.