Let be a finite G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B. It was observed that the quotient graph B of relative to B can be (G, 2)-arc transitive even if itself is not necessarily (G, 2)-arc transitive. In a previous article of Iranmanesh et al., this observation motivated a study of G-symmetric graphs ( , B) such that B is (G, 2)-arc transitive and, for blocks B, C ∈ B adjacent in B , there are exactly |B| − 2 (≥1) vertices in B which have neighbors in C. In the present article we investigate the general case where B is (G, 2)-arc transitive and is not multicovered by (i.e., at least one vertex in B has no neighbor in C for adjacent B, C ∈ B) by analyzing the dual D * (B) of the 1-design

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Let n ≥ 23 be an integer and let D 2n be the dihedral group of order 2n. It is proved that, if g 1 , g 2 , . . . , g 3n is a sequence of 3n elements in D 2n , then there exist 2n distinct indices i 1 , i 2 , . . . , i 2n such that g i 1 g i 2 · · · g i 2n = 1. This result is a sharpening of the famous Erdős-Ginzburg-Ziv theorem for G = D 2n .

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