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

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), ..., g2(p-1) of 2p-1 elements from G, the number of ways of writing a as a sum of exactly p of the gi'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.