PARTIAL ASSOCIATIVITY AND ROUGH APPROXIMATE GROUPS

Suppose that a binary operation. on a finite set X is injective in each variable separately and also associative. It is easy to prove that (X, omicron) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples (x, y, z) is an element of X-3 satisfy the equation x omicron (y omicron z) = (x omicron y) omicron z. Other results in additive combinatorics would lead one to expect that there must be an underlying 'group-like' structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.