Let G be an abelian group. For a subset A ⊂ G, denote by
2 ∧ A the set of sums of two different elements of A. A conjecture by
Erdős and Heilbronn, first proved by Dias da Silva
and Hamidoune, states that, when G has prime order,
[mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 2[mid ]A[mid ] − 3).
We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality
[mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 3[mid ]A[mid ]/2)
holds when A is a generating set of G, 0 ∈ A and
[mid ]A[mid ] [ges ] 21 (respectively, [mid ]A[mid ] [ges ] 33). The
structure of the sets for which equality holds is also determined.