Snevily's conjecture for subsets of size two
Suppose is an odd-order abelian group and and are (not necessarily disjoint) subsets of of size two. Then, one of these is true:
Given: A finite Abelian group of odd order, subsets and of .
To prove: Either or .
Proof: Suppose equality holds in both cases. Then, subtracting the two equations, we get:
Rearranging this, we get:
Since , , and since the group is Abelian of odd order, its double is also therefore nonzero (using fact (1)).