Sum-free subsets of finite abelian groups of type III

Abstract

A finite abelian group G of cardinality n is said to be of type III if every prime divisor of n is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group G of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group G. We supplement this result with a theorem on the structure of sum-free subsets of cardinality “close” to the largest possible in a type III abelian group G. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of Aut(G) on the set of sum-free subsets of G of the largest cardinality when G is of the form (Z/mZ)r, with all prime divisors of m congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of G. For finite abelian groups G of type III and with a given exponent this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa.