Improving the Erdős–Ginzburg–Ziv theorem for some non-abelian groups

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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 Erdős–Ginzburg–Ziv theorem gives the upper bound s ( G ) ⩽ 2 m − 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, Erdős–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.