The Chevalley-Warning theorem and a combinatorial question on finite groups

Abstract:

Recently, W. D. Gao (1996) proved the following theorem: For a cyclic group $G$ of prime order, and any element $a$ in it, and an arbitrary sequence $g_1, \ldots , g_{2p-1}$ of $2p-1$ elements from $G$, the number of ways of writing $a$ as a sum of exactly $p$ of the $g_i$’s is $1$ or $0$ modulo $p$ according as $a$ is zero or not. The dual purpose of this note is (i) to give an entirely different type of proof of this theorem; and (ii) to solve a conjecture of J. E. Olson (1976) by answering an analogous question affirmatively for solvable groups.