The Erdős–Ginzburg–Ziv theorem for dihedral groups

Let n≥23n≥23 be an integer and let D2nD2n be the dihedral group of order 2n2n. It is proved that, if g1,g2,…,g3ng1,g2,…,g3n is a sequence of 3n3n elements in D2nD2n, then there exist 2n2n distinct indices i1,i2,…,i2ni1,i2,…,i2n such that gi1gi2⋯gi2n=1gi1gi2⋯gi2n=1. This result is a sharpening of the famous Erdős–Ginzburg–Ziv theorem for G=D2nG=D2n.