Closest multiplication tables of groups

Abstract

Suppose that all groups of order $n$ are defined on the same set $G$ of cardinality $n$, and let the \emph{distance} of two groups of order $n$ be the number of pairs $(a,b)\in G\times G$ where the two group operations differ. Given a group $G(\circ )$ of order $n$, we find all groups of order $n$, up to isomorphism, that are closest to $G(\circ )$.