Restricted sumsets in a finite abelian group

In this paper, we Prove that if A and B are subsets of a finite abelian group G with vertical bar A vertical bar + vertical bar B vertical bar = vertical bar G vertical bar + L(G), then vertical bar A (+) over capB vertical bar >= vertical bar G vertical bar - 2, where L(G) = vertical bar{g : g is an element of G, 2g = 0}vertical bar and A (+) over capB = {a + b : a is an element of A, b is an element of B, a not equal b}. In addition, we give a complete description of the subsets A and B of G such that vertical bar A vertical bar + vertical bar B vertical bar = vertical bar G vertical bar + L(G) and A (+) over cap B not equal G. Our results generalize the corresponding theorems of Gallardo et al. in cyclic group Z/nZ [L. Gallardo, G. Grekos, L Habsieger, et al., Restricted addition in Z/nZ and an application to the Erdos-Ginzburg-Ziv problem, J. London Math. Soc. 65 (2) (2002) 513-523]. (C) 2009 Elsevier B.V. All rights reserved.