On The Critical Number And Its Inverse Problem

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. If S is an additive basis of G, we also call it an complete set. The critical number cγ(G) be the smallest integer t such that every subset of G\{0} of cardinality t is an additive basis of G. To determine cγ(G) of finite groups is an important topic of additive combinatorics which deals with combinatorial properties of algebraic objects, typically abelian groups, rings, or fields.The study of the parameter cγ(G) stems from the 1964 work of Erdos and Heilbronn on the case G=Zp. They showed that if S is a set of non-zero elements of Zp with |S|≥3(?), then the subset sums of S, together with 0, cover Zp. This was improved by Olson to cγ(Zp)≤(?)+1. Much later, in 1994, Dias da Silva and Hamidoune obtained crγ(Zp)≤(?) which is essentially best possible. The evaluation of cγ(G) for groups with composite order was first considered in 1967 by Mann and Olson. They obtained the inequality cγ(Zp Zp)≤2p-1. Mann and Wou obtained cγ(Zp⊕Zp)=2p-2. Since then, with many mathematicians' efforts, critical number has been determined for all finite abelian groups recently.However, the problem to determine critical number for non-abelian groups is widely open. So far, we only have a few results obtained by H.B. Mann, J.E. Olson, and W. Gao. Combining the methods from Alon, Gao, Hamidoune and Diderrich we will determine the critical number of finite nilpotent groups in sec-tion two. We also would like to study the inverse problem of critical number. In section three, we obtained characterizations of incomplete sets of abelian group of order pq. In section four, using Vu's method we get characterizations of in-complete sets of finite nilpotent groups.