  Let G be an additive abelian group whose finite subgroups are all cyclic. Let
A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let
b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for
every positive integer m\leq (k-1)/(n-1) there are more than
(k-1)n-(m+1)n(n-1)/2 sets {a_1,...,a_n} such that a_1\in A_1,..., a_n\in A_n,
and both a_i\not=a_j and ma_i+b_i\not=ma_j+b_j (or both ma_i\not=ma_j and
a_i+b_i\not=a_j+b_j) for all 1\leq i<j\leq n.
  This extends a recent result of Dasgupta, K\'arolyi, Serra and
  Szegedy on Snevily's conjecture. Actually stronger results on sumsets with
polynomial restrictions are obtained in this paper.
