On the distance between non-isomorphic groups

A result of Ben-Or, Coppersmith, Luby and Rubinfeld on testing whether a map between two groups is close to a homomorphism implies a tight lower bound on the distance between the multiplication tables of two non-isomorphic groups. In [2] Drápal showed that if ◦ and ∗ are two binary operations on the finite set G such that (G,◦) and (G,∗) are non-isomorphic groups then the Hamming distance between the two multiplication tables is greater than 1 9 |G| 2. In [3] infinite families of non-isomorphic pairs of 3-groups with distance exactly 9 |G| 2 are given. In this note we show that 9 |G| 2 is a lower bound for the distance of arbitrary non-isomorphic group structures. The proof is a simple application of the following result from [1]. Fact 1. Let (G,◦) and (K,∗) be two groups and f :G → K be a map such that #{(x,y) ∈ G×G : f (x◦ y) = f (x)∗ f (y)} |G|2 > 7 9 . Then there exists a group homomorphism h:G → K such that #{x∈G: f (x)=h(x)} |G| ≥ 5 9 . Fact 1 is a weak version of Theorem 1 in [1]. Here is a brief sketch of its proof. For every x ∈ G, h(x) is defined as the value taken most frequently by the expression f (x◦ y)∗ f (y)−1 where y runs over G. Then the first step is showing that for every x ∈ G, #{y ∈ G : f (x◦y)∗ f (y)−1 = h(x)} > 3 |G|. The homomorphic property of h and equality of h(x) with f (x) for 5 9 of the possible elements x follow from this claim easily. We apply Fact 1 to obtain a result on the distance of multiplication tables of groups of not necessarily equal size. It will be convenient to state it in terms of a quantity complementary to the distance. Let (G,◦) and (K,∗) be finite groups. We define the overlap between (G,◦) and (K,∗) as max γ :G→֒S,κ:K→֒S # 