Suppose that a binary operation \(\circ \) on a finite set X is injective in each variable separately and also associative. It is easy to prove that \((X,\circ )\) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples \((x,y,z)\in X^3\) satisfy the equation \(x\circ (y\circ z)=(x\circ y)\circ z\). 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.

假设对有限集X的二元运算\（\ circ \）在每个变量中分别是内射的，并且也是关联的。很容易证明\（（X，\ circ）\）必须是一个组。在本文中，我们研究了如果仅知道三元组\（（x，y，z）\在X ^ 3 \中）满足方程\（x \ circ（y \ circ z）=（ x \ circ y）\ circ z \）。加性组合法的其他结果会使人们期望，必须有一个潜在的“类群”结构负责大量的关联三元组。我们证明确实是这样：乘法表中必须有一个比例大小的子集，该子集与度量标准组的乘法表的一部分大致相符。Green的最新结果表明，这种度量近似是必要的：并非总能获得与组的乘法表的一部分相符的比例大小的子集。