Closest multiplication tables of groups

TL;DR

This paper investigates the problem of identifying all groups of a given order that are closest to a specified group, based on the minimal number of differing operation pairs, thereby exploring the structure of near-isomorphic groups.

Contribution

It provides a method to find all groups of a fixed order that are closest to a given group in terms of operation differences, extending understanding of group proximity.

Findings

Characterization of closest groups for various orders

Algorithms for computing minimal differences between groups

Insights into the structure of nearly identical 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 $G(\circ)$.