The Erdős–Ginzburg–Ziv theorem for finite solvable groups

Let GG be a non-cyclic finite solvable group of order nn, and let S=(g1,…,gk)S=(g1,…,gk) be a sequence of kk elements (repetition allowed) in GG. In this paper we prove that if k≥74n−1, then there exist some distinct indices i1,i2,…,ini1,i2,…,in such that the product gi1gi2⋯gin=1gi1gi2⋯gin=1. This result substantially improves the Erdős–Ginzburg–Ziv theorem and other existing results.