On Additive Bases

The critical number of G, denoted by c(G), is the smallest s such that Σ(S) = G for every subset S of G with cardinality s not containing 0. The parameter c(G) was first studied by Erdős and Heilbronn in [4]. They obtained the inequality c(Zp) ≤ 3 √ 6p. Olson proved in [13] that c(Zp) ≤ √ 4p− 3 + 1. The authors of [1] obtained the inequality c(Zp) ≤ √ 4p− 7. The evaluation of c(G) for groups with composite order was first considered by Mann and Olson. They obtained the inequality c(Zp⊕Zp) ≤ 2p− 1 in [11]. Mann and Wou proved that c(Zp ⊕ Zp) = 2p − 2 in [12]. Diderrich proved in [2] the inequality p + q − 2 ≤ c(G) ≤ p+ q − 1, where G is an abelian group of order pq and q is a prime. He conjectured that c(G) = |G|/p + p − 2 if |G|/p is composite, where p is the smallest prime dividing |G|. This conjecture is proved by Diderrich and Mann in [3] for p = 2. Peng [15] proved Diderrich’s conjecture if G is the additive group of a finite field. Lipkin [9] obtained a proof of this conjecture in the case of cyclic groups with large order. This conjecture is proved by one of the present authors in [5] for p ≥ 43 and by the authors of [8] for p = 3. In this paper we achieve the evaluation of c(G), solving the above mentioned conjecture.