Abstract <p> A permutation group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> of a finite set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega\)</EquationSource> </InlineEquation> acts componentwise on the Cartesian square <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega^2\)</EquationSource> </InlineEquation>. The largest subgroup of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\operatorname{\mathrm{Sym}}(\Omega)\)</EquationSource> </InlineEquation> that has the same orbits on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Omega^2\)</EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is called the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation>-closure of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation>. The rank of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is the number of its orbits on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\Omega^2\)</EquationSource> </InlineEquation>. If the rank of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation> and the order is even, then an undirected graph with vertex set <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\Omega\)</EquationSource> </InlineEquation> is defined, up to taking the complement, for which one of the two non-diagonal orbits of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> on <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(\Omega^2\)</EquationSource> </InlineEquation> is taken as the edge set. Such a graph is called a rank <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation> graph. The full automorphism group of this graph coincides with the <InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation>-closure of <InlineEquation ID="IEq18"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> and contains <InlineEquation ID="IEq19"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> as a subgroup. At present, with the exception of the case when <InlineEquation ID="IEq20"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> is an almost simple group, there exists an explicit description of the <InlineEquation ID="IEq21"> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation>-closures of rank <InlineEquation ID="IEq22"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation> groups <InlineEquation ID="IEq23"> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation>. In this paper, we fill the existing gap, thereby completing the description of the automorphism groups of rank <InlineEquation ID="IEq24"> <EquationSource Format="TEX">\(3\)</EquationSource> </InlineEquation> graphs. </p>

On the Almost Simple Automorphism Groups of Rank 3 Graphs

期刊论文
发表日期：

2025年8月1日

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Z. Wang,
A. V. Vasil’ev,
D. O. Revin

摘要

Abstract

A permutation group \(G\) of a finite set \(\Omega\) acts componentwise on the Cartesian square \(\Omega^2\) . The largest subgroup of \(\operatorname{\mathrm{Sym}}(\Omega)\) that has the same orbits on \(\Omega^2\) as \(G\) is called the \(2\) -closure of \(G\) . The rank of \(G\) is the number of its orbits on \(\Omega^2\) . If the rank of \(G\) is \(3\) and the order is even, then an undirected graph with vertex set \(\Omega\) is defined, up to taking the complement, for which one of the two non-diagonal orbits of \(G\) on \(\Omega^2\) is taken as the edge set. Such a graph is called a rank \(3\) graph. The full automorphism group of this graph coincides with the \(2\) -closure of \(G\) and contains \(G\) as a subgroup. At present, with the exception of the case when \(G\) is an almost simple group, there exists an explicit description of the \(2\) -closures of rank \(3\) groups \(G\) . In this paper, we fill the existing gap, thereby completing the description of the automorphism groups of rank \(3\) graphs.