Abstract We survey the state of research to determine the maximum size of a nonspanning subset of a finite abelian group G of order n. The smallest prime factor of n, denote it here by p, plays a crucial role. For prime order, G= Z p , this is essentially an old problem of Erdős and Heilbronn, which can be solved using a result of Dias da Silva and Hamidoune. We provide a simple new proof for the solution when n is even (p=2). For composite odd n, we deduce the solution, for n⩾2p2, from results obtained years ago by Diderrich and, recently, by Gao and Hamidoune. Only a small family of cases remains unsettled.
[1]
John E. Olson,et al.
An addition theorem modulo p
,
1968
.
[2]
G. Diderrich,et al.
An addition theorem for Abelian groups of order pq
,
1975
.
[3]
Noga Alon,et al.
The Polynomial Method and Restricted Sums of Congruence Classes
,
1996
.
[4]
J. A. Dias da Silva,et al.
Cyclic Spaces for Grassmann Derivatives and Additive Theory
,
1994
.
[5]
R. Guy.
Unsolved Problems in Number Theory
,
1981
.
[6]
H. B. Mann,et al.
An addition theorem for the elementary Abelian group of type (p, p)
,
1986
.