Finite Permutation Groups Research Articles

On the regularity number of a finite group and other base‐related invariants

AbstractA ‐tuple of core‐free subgroups of a finite group is said to be regular if has a regular orbit on the Cartesian product . The regularity number of , denoted by , is the smallest positive integer with the property that every such ‐tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that and . We also formulate and investigate natural generalisations of several well‐studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that for every almost simple sporadic group, with equality if and only if is the Mathieu group . We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.

Rigid stabilizers and local prosolubility for boundary-transitive actions on tree

Abstract Let 𝐺 be a group acting 2-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where 𝐺 has both P ⁢ Γ ⁢ L 3 ⁢ ( 4 ) \mathrm{P}\Gamma\mathrm{L}_{3}(4) and P ⁢ Γ ⁢ L 3 ⁢ ( 5 ) \mathrm{P}\Gamma\mathrm{L}_{3}(5) as local actions. We show that, for each half-tree T a T_{a} , the local action of the rigid stabilizer of T a T_{a} at the root of T a T_{a} contains the soluble residual of the point stabilizer of the local action of 𝐺. In particular, 𝐺 is locally prosoluble if and only if its local actions have soluble point stabilizers; if 𝐺 is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in 𝐺. These results are partly inspired by Radu’s classification of groups acting boundary-2-transitively on trees with local action containing the alternating group, and partly based on the author’s recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks.

On the cardinality of irredundant and minimal bases of finite permutation groups

Given a finite permutation group G with domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}, we associate two subsets of natural numbers to G, namely I(G,Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {I}}(G,\\Omega )$$\\end{document} and M(G,Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}(G,\\Omega )$$\\end{document}, which are the sets of cardinalities of all the irredundant and minimal bases of G, respectively. We prove that I(G,Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {I}}(G,\\Omega )$$\\end{document} is an interval of natural numbers, whereas M(G,Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}(G,\\Omega )$$\\end{document} may not necessarily form an interval. Moreover, for a given subset of natural numbers X⊆N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X \\subseteq {\\mathbb {N}}$$\\end{document}, we provide some conditions on X that ensure the existence of both intransitive and transitive groups G such that I(G,Ω)=X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {I}}(G,\\Omega ) = X$$\\end{document} and M(G,Ω)=X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}(G,\\Omega ) = X$$\\end{document}.

Spreading primitive groups of diagonal type do not exist

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between$2$-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

An Interview with Professor Rosemary Bailey

Professor Rosemary A. Bailey is a giant in the field of applied statistics and design of experiments. She received her undergraduate and D.Phil degrees at the University of Oxford. Her 1974 D.Phil. dissertation was on finite permutation groups supervised by Professor Graham Higman [1917-2008) one of the three most significant British group theorists of the 20th century. She is Professor Emerita of Statistics at Queen Mary, University of London, and currently Professor of Mathematics and Statistics at the University of St Andrews, Scotland.

Vector invariants of permutation groups in characteristic zero

We consider a finite permutation group acting naturally on a vector space [Formula: see text] over a field [Formula: see text]. A well-known theorem of Göbel asserts that the corresponding ring of invariants [Formula: see text] is generated by the invariants of degree at most [Formula: see text]. In this paper, we show that if the characteristic of [Formula: see text] is zero, then the top degree of vector coinvariants [Formula: see text] is also bounded above by [Formula: see text], which implies the degree bound [Formula: see text] for the ring of vector invariants [Formula: see text]. So, Göbel’s bound almost holds for vector invariants in characteristic zero as well.

Computing canonical images in permutation groups with Graph Backtracking

We describe a new algorithm for finding a canonical image of an object under the action of a finite permutation group. This algorithm builds on previous work using Graph Backtracking [9], which extends Jeffrey Leon's Partition Backtrack framework [14,15]. Our methods generalise both Nauty [17] and Steve Linton's Minimal image algorithm [16].

New characterizations of primitive permutation groups with applications to synchronizing automata

For a finite permutation group on n elements we show the following (and variants thereof) equivalences: (1) the permutation group is primitive, (2) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every non-empty subset, an element mapping the whole permutation domain onto this subset, (3) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every two distinct subsets, an element mapping precisely one to a singleton set. We also investigate further properties related to the reachability of subsets. Lastly, we apply our results to automata and show that automata whose transformation monoids contain a primitive permutation group and a mapping that excludes precisely one state from its image are completely reachable and have the property that a minimal automaton for the set of synchronizing words has the maximal possible number of states.

IBIS soluble linear groups

Let G be a finite permutation group on An ordered sequence of elements of Ω is an irredundant base for G if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of G have the same cardinality, G is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.

Class-two quotients of finite permutation groups

Abstract Let 𝐺 be a permutation group on a finite set and let 𝑝 be a prime. In this paper, we prove that the largest class-two 𝑝-quotient of 𝐺 has order at most p n / p p^{n/p} (or 2 3 ⁢ n / 4 2^{3n/4} if p = 2 p=2 ), where 𝑛 is the number of points moved by a Sylow 𝑝-subgroup of 𝐺. Further, we describe the groups whose largest class-two 𝑝-quotients can reach such a bound. This extends earlier work of Kovács and Praeger from 1989.

Two-closure of rank 3 groups in polynomial time

A finite permutation group G on Ω is a rank 3 group if it has precisely three orbits in its induced action on Ω×Ω. The largest permutation group on Ω having the same orbits as G on Ω×Ω is the 2-closure of G. We construct a polynomial-time algorithm which, given generators of a rank 3 group, computes generators of its 2-closure.

Totally 2-closed finite groups with trivial Fitting subgroup

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

On base sizes for primitive groups of product type

Let G ⩽ Sym ( Ω ) be a finite permutation group and recall that the base size of G is the minimal size of a subset of Ω with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form L ≀ P ⩽ Sym ( Ω ) with soluble point stabilisers, where Ω = Γ k , L ⩽ Sym ( Γ ) and P ⩽ S k is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form L ≀ P and we classify the groups with a unique regular suborbit under the assumption that P is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups.

A classification of finite primitive IBIS groups with alternating socle

Abstract Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , … , ω ℓ ) (\omega_{1},\ldots,\omega_{\ell}) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups 𝐺 to the case that the socle of 𝐺 is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

Index graphs of finite permutation groups

Let <em>G</em> be a subgroup of <em>S</em><sub>n</sub>. For <em>x ∈ G</em>, the index of <em>x</em> in <em>G</em> is denoted by <em>ind x</em> is the minimal number of 2-cycles needed to express <em>x</em> as a product. In this paper, we define a new kind of graph on <em>G</em>, namely the index graph and denoted by <em>Γ</em><sup>ind</sup><em>(G)</em>. Its vertex set the set of all conjugacy classes of <em>G</em> and two distinct vertices <em>x ∈ C</em><sub>x</sub> and <em>y ∈ C</em><sub>y</sub> are adjacent if <em>Gcd(ind x, ind y) 6 ≠ 1</em>. We study some properties of this graph for the symmetric groups <em>S</em><sub>n</sub>, the alternating group <em>A</em><sub>n</sub>, the cyclic group <em>C</em><sub>n</sub>, the dihedral group <em>D</em><sub>2n</sub> and the generalized quaternain group <em>Q</em><sub>4n</sub>. In particular, we are interested in the connectedness of them.

De Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products

Local actions of [Formula: see text], the group of finite permutations on [Formula: see text], on quasi-local algebras are defined and proved to be [Formula: see text]-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of [Formula: see text]-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon [Formula: see text] in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here.

Closures of Finite Permutation Groups

Closures of Finite Permutation Groups

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.

Permute and Add Network Codes via Group Algebras

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as <i>permute-and-add</i> network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their <i>degree</i> which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature.

Perfect refiners for permutation group backtracking algorithms

Backtrack search is a fundamental technique for computing with finite permutation groups, which has been formulated in terms of points, ordered partitions, and graphs. We provide a framework for discussing the most common forms of backtrack search in a generic way. We introduce the concept of perfect refiners to better understand and compare the pruning power available in these different settings. We also present a new formulation of backtrack search, which allows the use of graphs with additional vertices, and which is implemented in the software package Vole. For each setting, we classify the groups and cosets for which there exist perfect refiners. Moreover, we describe perfect refiners for many naturally-occurring examples of stabilisers and transporter sets, including applications to normaliser and subgroup conjugacy problems for 2-closed groups.