The Erdős–Ginzburg–Ziv theorem for finite solvable groups

Abstract

Let G be a non-cyclic finite solvable group of order n , and let S = ( g 1 , … , g k ) be a sequence of k elements (repetition allowed) in G . In this paper we prove that if k ≥ 7 4 n − 1 , then there exist some distinct indices i 1 , i 2 , … , i n such that the product g i 1 g i 2 ⋯ g i n = 1 . This result substantially improves the Erdős–Ginzburg–Ziv theorem and other existing results.