How far apart can the group multiplication tables be? (Q1199770)

Suppose \(G\) is a set of cardinality \(n \geq 2\) and two group structures \(G(\cdot)\) and \(G(*)\) are defined on \(G\). The Hamming distance of these two group structures is defined by \(\text{dist}(G(\cdot),G(*)) = \text{card}\{(a,b) \in G \times G\mid a\cdot b \neq a*b\}\) and for any group \(G(\cdot)\) define \(\delta(G(\cdot)) = \text{min }\text{dist}(G(\cdot),G(*))\) with \(G(*)\) running through all group structures on \(G\) such that \(G(*) \neq G(\cdot)\). In the paper under review the author proves that for \(n \geq 51\) \(\delta(G(\cdot))\) is one of the numbers \(6n - 24\), \(6n - 20\) or \(6n - 18\). Furthermore he proves that if \(\text{dist}(G(\cdot),G(*)) < n^ 2/9\), then the groups \(G(\cdot)\) and \(G(*)\) are isomorphic.