In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions X, S and E (defined in the case of N) and obtain bounds on them. Our estimates on the functions SG and EG are valid for general abelian groups G while in the case of XG we show that distinct types of behaviours may occur depending on G.

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Let n and k be integers. A set A ⊂ Z/nZ is k-free if for all x in A, kx / ∈ A. We determine the maximal cardinality of such a set when k and n are coprime. We also study several particular cases and we propose an efficient algorithm for solving the general case. We finally give the asymptotic behaviour of the minimal size of a k-free set in 1, n which is maximal for inclusion.