Improving the Erdos-Ginzburg-Ziv theorem for some non-abelian groups

Let G be a group of order m. Define s(G) to be the smallest value of t such that out of any t elements in G, there are m with product 1. The Erdos-Ginzburg-Ziv theorem gives the upper bound s(G) <= 2m - 1, and a lower bound is given by s(G) >= D(G) +m - 1, where D(G) is Davenport's constant. A conjecture by Zhuang and Gao [J.J. Zhuang, W.D. Gao, Erdos-Ginzburg-Ziv theorem for dihedral groups of large prime index, European J. Combin. 26 (2005) 1053-1059] asserts that s(G) = D(G) +m - 1, and Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103] has proven this for all abelian G. In this paper we verify the conjecture for a few classes of non-abelian groups: dihedral and dicyclic groups, and all non-abelian groups of order pq for p and q prime. (C) 2007 Elsevier Inc. All rights reserved.