Finite Nonabelian Simple Group Research Articles

On the Uniqueness Theorem

The aim of this paper is to investigate whether the Uniqueness Theorem introduced by Feit and Thompson holds in finite simple groups of Lie type of Lie rank one. We also introduce and examine the Strong Uniqueness Theorem. We define the concept of containment minimality and prove that a non-Abelian finite simple group is minimal if and only if it is containment minimal. Finally, we inquire when a minimal non-soluble group has a faithful doubly transitive representation.

Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Let [Formula: see text] be a finite group and [Formula: see text] the set of all irreducible complex characters of [Formula: see text]. Let [Formula: see text] be the set of all irreducible complex character degrees of [Formula: see text] and denote by [Formula: see text] the set of all primes which divide some character degree of [Formula: see text]. The character-prime graph [Formula: see text] associated to [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and there is an edge between two distinct primes [Formula: see text] and [Formula: see text] if and only if the product [Formula: see text] divides some character degree of [Formula: see text]. We show that the finite nonabelian simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are uniquely determined by their degree-patterns and orders.

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.

Involutions in finite simple groups as products of conjugates

Let G be a finite non-abelian simple group, C a non-identity conjugacy class of G, and Γ C the Cayley graph of G based on C ∪ C − 1 . Our main result shows that in any such graph, there is an involution at bounded distance from the identity.

Connectivity of the reduced power graphs of finite simple groups

The reduced power graph of a group G is a graph whose vertices are the non-identity elements of G, with two vertices connected by an edge if and only if one is a power of the other. Akbari and Ashrafi conjectured that if a non-abelian finite simple group has a connected reduced power graph, then it must be an alternating group. In this paper, we disprove this conjecture by providing a complete description of when the reduced power graphs of PGLn(Fq) are connected for all q and all n≥3. We also provide upper bounds on their diameters and, in case of disconnection, a description of all connected components. Our results have implications for the study of power graphs of other families of groups.

Hypermaps Over Non-Abelian Simple Groups and Strongly Symmetric Generating Sets

A generating pair $x, y$ for a group $G$ is said to be symmetric if there exists an automorphism $\varphi_{x,y}$ of $G$ inverting both $x$ and $y$, that is, $x^{\varphi_{x,y}}=x^{-1}$ and $y^{\varphi_{x,y}}=y^{-1}$. Similarly, a group $G$ is said to be strongly symmetric if $G$ can be generated with two elements and if all generating pairs of $G$ are symmetric.
 In this paper we classify the finite strongly symmetric non-abelian simple groups. Combinatorially, these are the finite non-abelian simple groups $G$ such that every orientably regular hypermap with monodromy group $G$ is reflexible.

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.

Total closure for permutation actions of finite nonabelian simple groups

For a positive integer k, a group G is said to be totally k-closed if for each set 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} upon which G acts faithfully, G is the largest subgroup of Sym(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sym}(\\Omega )$$\\end{document} that leaves invariant each of the G-orbits in the induced action on Ω×⋯×Ω=Ωk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \ imes \\cdots \ imes \\Omega =\\Omega ^k$$\\end{document}. Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. We address the question of determining the closure number k(G) for finite simple groups G. Prior to our work it was known that k(G)=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(G)=2$$\\end{document} for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that k(G)⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(G)\\geqslant 3$$\\end{document} for all other finite simple groups. We determine the value for the alternating groups, namely k(An)=n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(A_n)=n-1$$\\end{document}. In addition, for all simple groups G, other than alternating groups and classical groups, we show that k(G)⩽7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(G)\\leqslant 7$$\\end{document}. Finally, if G is a finite simple classical group with natural module of dimension n, we show that k(G)⩽n+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(G)\\leqslant n+2$$\\end{document} if n⩾14\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\geqslant 14$$\\end{document}, and k(G)⩽⌊n/3+12⌋\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(G) \\leqslant \\lfloor n/3 + 12 \\rfloor $$\\end{document} otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on n and the type of G) on the base sizes of the primitive actions of G, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.

Projective Indecomposable Permutation Modules

We investigate finite non-Abelian simple groups G for which the projective cover of the trivial module coincides with the permutation module on a subgroup and classify all cases unless G is of Lie type in defining characteristic.

On continuous orbit equivalence rigidity for virtually cyclic group actions

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group by certain other virtually cyclic groups, e.g., the direct product of the integer group with any non-abelian finite simple group.

The isomorphism of generalized Cayley graphs on finite non-abelian simple groups

The isomorphism problem is a fundamental problem for algebraic and combinatorial structures, particularly in relation to Cayley graphs. Let Xi=GC(G,Si,αi),(i=1,2) be generalized Cayley graphs. If whenever X1≅X2, it implies that α2=α1γ and S2=g−1S1γgα2 for some g∈G and γ∈Aut(G), then G is a strongly generalized Cayley isomorphism (GCI)-group. In this study, we defined (strongly, restricted) m-GCI-groups. These definitions are similar to those of m-CI-groups for Cayley graphs. Our main results demonstrate that a finite non-abelian simple group G is a restricted 2-GCI-group if and only if G is one of A5, L2(8), M11, Sz(8), or M23, and G is a 2-GCI-group if and only if G is A5 or L2(8).

Characteristic covering numbers of finite simple groups

In the past few decades there has been considerable interest in word maps on groups with emphasis on (non-abelian) finite simple groups. Various asymptotic results (holding for sufficiently large groups) have been obtained. More recently non-asymptotic results (holding for all finite simple groups) emerged, with emphasis on particular words (commutators and certain power words) which are not an identity of any finite simple group. In this paper we initiate a systematic study of all words with the above property. In particular, we show that, if \(w_1, \ldots , w_6\) are words which are not an identity of any (non-abelian) finite simple group, then \(w_1(G)w_2(G) \ldots w_6(G) = G\) for all (non-abelian) finite simple groups G. Consequently, for every word w, either \(w(G)^6 = G\) for all finite simple groups, or \(w(G)=1\) for some finite simple group. These theorems follow from more general results we obtain on characteristic collections of finite groups and their covering numbers, which are of independent interest and have additional applications.

Counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky

In this article, we provide counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky which states that each coset of the centralizer of an involution in a finite non-abelian simple group $G$ contains an odd order element, unless $G=\text{PSL}(n,2)$ for $n\geq 4$. More precisely, we show that the conjecture does not hold for the alternating group $A_{8n}$ for all $n\geq 2$.

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.

On recognizing groups by the bottom layer

The article discusses the possibility of recognizing a group by the bottom layer, that is, by the set of its elements of prime orders. The paper gives examples of groups recognizable by the bottom layer, almost recognizable by the bottom layer, and unrecognizable by the bottom layer. Results are obtained for recognizing a group by the bottom layer in the class of infinite groups under some additional restrictions. The notion of recognizability of a group by the bottom layer was introduced by analogy with the recognizability of a group by its spectrum (the set of orders of its elements). It is proved that all finite simple nonAbelian groups are recognizable by spectrum and bottom layer simultaneously in the class of finite simple non-Abelian groups.

Cubic graphical regular representations of some classical simple groups

A graphical regular representation (GRR) of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. In this paper we study cubic GRRs of PSLn(q) (n=4,6,8), PSpn(q) (n=6,8), PΩn+(q) (n=8,10,12) and PΩn−(q) (n=8,10,12), where q=2f with f≥1. We prove that for each of these groups, with probability tending to 1 as q→∞, any element x of odd prime order dividing 2ef−1 but not 2i−1 for each 1≤i<ef together with a random involution y gives rise to a cubic GRR, where e=n−2 for PΩn+(q) and e=n for other groups. Moreover, for sufficiently large q, there are elements x satisfying these conditions, and for each of them there exists an involution y such that {x,x−1,y} produces a cubic GRR. This result together with certain known results in the literature implies that except for PSL2(q), PSL3(q), PSU3(q) and a finite number of other cases, every finite non-abelian simple group contains an element x and an involution y such that {x,x−1,y} produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.

The diameter of products of finite simple groups

Following partially a suggestion by Pyber, we prove that the diameter of a product of non-abelian finite simple groups is bounded linearly by the maximum diameter of its factors. For completeness, we include the case of abelian factors and give explicit constants in all bounds.

Jordan decomposition for weights and the blockwise Alperin weight conjecture

The Alperin weight conjecture was reduced to simple groups by the work of Navarro–Tiep and Späth. To prove Alperin weight conjecture, it suffices to show that all finite non-abelian simple groups are BAW-good. We reduce the verification of the inductive conditions for groups of Lie type in non-defining characteristic to quasi-isolated blocks.

Applications of group theory in crystallography

Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an overview of the results on their recognition by the spectrum (by the orders of the elements of the group). This direction has been intensively developed in recent years both in our country and abroad. Recognition of finite simple non-Abelian groups by spectrum has been studied for last thirty years in Yekaterinburg at the Institute of Mathematics and Mechanics of the Ural Division of the Russian Academy of Sciences, in Chelyabinsk Federal University and in the Novosibirsk Institute of Mathematics of Siberian Division of the Russian Academy of Sciences. Some simple non-Abelian groups are not recognizable by their spectra. We have proposed an approach for recognizing groups by the bottom layer. The bottom layer of a group is the set of its elements of prime orders. A group is called recognizable by the bottom layer under additional conditions if it is uniquely restored by the bottom layer under these conditions. The paper considers some examples of simple non-Abelian finite groups that are not recognizable by spectra. For these examples, simultaneous recognition by spectrum and by the bottom layer is proved.

The time complexity of some algorithms for generating the spectra of finite simple groups

The spectrum $\omega(G)$ is the set of orders of elements of $G$. We consider the problem of generating the spectrum of a finite nonabelian simple group $G$ given by the degree of $G$ if $G$ is an alternating group, or the Lie type, Lie rank and order of the underlying field if $G$ is a group of Lie type.