# Snevily's conjecture for subsets of size two

## Contents

## Statement

Suppose is an odd-order abelian group and and are (not necessarily disjoint) subsets of of size two. Then, one of these is true:

.

## Related facts

- Snevily's conjecture: A related question for subsets of bigger size is an open conjecture.
- Snevily's conjecture for cyclic groups

## Facts used

## Proof

**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)).