Sum-free subsets of finite abelian groups of type III

Details Sum-free subsets of finite abelian groups of type III Sum-free subsets of finite abelian groups of type III

2007-11-27

arxiv.org/abs/0711.4317v3

Mathematics

Number Theory

24 pages, no figure, a reference number corrected

Scientific paper

A finite abelian group $G$ of order $n$ is said to be of type III if all divisors of $n$ are congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of cardinality very "close" to the largest possible 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 then give two applications of this theorem. Our first application allows us to write down a formula for the number of orbits under the natural action of ${\rm Aut}(G)$ on the set of sum-free subsets of $G$ of the largest cardinality when $G$ is of the form $ (Z/mZ)^r$, with all 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.

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