Bounds for counter-examples to addition theorems in solvable groups

Abstract

Introduction. It is well known that any set of 2 n - i elements of a solvable group of size n must contain a subset of size n with the property that the product of the dements of this subset arranged in the appropriate order yields the identity. In fact, if G is not cyclic, it is sufficient that the original set contain 2 n - 2 elements. The main result of this paper is that similar results hold for sets of size 2 n - r, where r is a fixed positive integer. The result says that for a fixed r, if n is sufficiently large and G is a non-cyclic solvable group, then any set of 2 n - r elements of G contains a subset of size n with the property that the product of the elements of this subset in the appropriate order is the identity. The term "set" in this paper actually means multi-set. That is, an element may occur more than once in a set. The cardinality of that set is number of elements in the set counting multiplicities. When we wish to consider an object which is a set in the classical sense, we shall use the term "ordinary set", All of the results in this paper concern sets of elements from a finite group. In some cases, the group is assumed to be abelian. In these cases, we shall use additive notation. In all other cases, multiplicative notation will be used. D e f i n i t i o n. Let G be a group and let S be a set of elements of G. A n-sum in S is an ordered subset of S of cardinality n. The result of that n-sum is the product of the elements in the n-sum in the specified order.