Classification Of Simple Groups Research Articles

Diameter of classical groups generated by transvections

Let G be a finite classical group generated by transvections, i.e., one of SLn(q), SUn(q), Sp2n(q), or O2n±(q)(qeven), and let X be a generating set for G containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph Cay(G,X) is bounded by (nlog⁡q)C for some constant C. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection.By combining this with a result of the author and Jezernik it follows that if G is one of SLn(q), SUn(q), Sp2n(q) and X contains three random generators then with high probability the diameter Cay(G,X) is bounded by nO(log⁡q). This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators.

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

Totaro’s question for adjoint groups of types 𝐴₁ and 𝐴_{2𝑛}

Let G G be a smooth connected linear algebraic group over a field k k , and let X X be a G G -torsor. Totaro asked: if X X admits a zero-cycle of degree d ≥ 1 d \geq 1 , then does X X have a closed étalé point of degree dividing d d ? We give an affirmative answer for absolutely simple classical adjoint groups of types A 1 A_1 and A 2 n A_{2n} over fields of characteristic ≠ 2 \neq 2 .

Generating functions for intersection products of divisors in resolved F-theory models

Building on the approach of 1703.00905, we present an efficient algorithm for computing topological intersection numbers of divisors in a broad class of elliptic fibrations with the aid of a symbolic computing tool. A key part of our strategy is organizing the intersection products of divisors into a succinct analytic generating function, namely the exponential of the Kähler class. We use the methods of 1703.00905 to compute the pushforward of this function to the base of the elliptic fibration. We implement our algorithm in an accompanying Mathematica package IntersectionNumbers.m that computes generating functions of intersection products for resolutions of F-theory Tate models defined over smooth base of arbitrary complex dimension. Our algorithm appears to offer a significant reduction in computation time needed to compute intersection numbers as compared to previously explored implementations of the methods in 1703.00905; as an illustration, we explicitly compute the generating functions for all F-theory Tate models with simple classical groups of rank up to twenty and highlight the growth of the computation time with the rank of the group.

ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

In the present paper, we classify abelian antipodal distance-regular graphs \(\Gamma\) of diameter 3 with the following property: \((*)\) \(\Gamma\) has a transitive group of automorphisms \(\widetilde{G}\) that induces a primitive almost simple permutation group \(\widetilde{G}^{\Sigma}\) on the set \({\Sigma}\) of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank \({\rm rk}(\widetilde{G}^{\Sigma})\) of \(\widetilde{G}^{\Sigma}\) equals 2 moreover, all such graphs are now known. Here we focus on the case \({\rm rk}(\widetilde{G}^{\Sigma})=3\).Under this condition the socle of \(\widetilde{G}^{\Sigma}\) turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs \(\Gamma\) with the property \((*)\) such that \(rk(\widetilde{G}^{\Sigma})=3\) and the socle of \(\widetilde{G}^{\Sigma}\) is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for \(\widetilde{G}^{\Sigma}\). We follow a classification scheme that is based on a reduction to minimal quotients of \(\Gamma\) that inherit the property \((*)\). For each given group \(\widetilde{G}^{\Sigma}\) with simple classical socle of degree \(|{\Sigma}|\le 2500\), we determine potential minimal quotients of \(\Gamma\), applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of \(\Gamma\) in the case of classical socle for \(\widetilde{G}^{\Sigma}\) under condition \(|{\Sigma}|\le 2500.\)

The Significance of Cyclic Groups

Investigation in group theory has been a widespread practice within the realm of mathematics. In finite group theory specifically, mathematicians have historically aimed to classify finite simple groups. Following the turning consensus that the finite simple group classification is complete, research within finite group theory has shifted to other areas. However, this paper endeavors to fill the missing gaps in the current classification – it does not specify the importance of finite group theory. By focusing on introducing, explaining, and demonstrating cyclic groups and their applications, the paper will reveal, specifically, the importance of cyclic groups in the realm of mathematics and beyond. In particular, the qualities of finite cyclic groups allow them to apply easily and broadly from as simple as division rules and binary number systems (by use of a set of integers of modulus n), to more advanced theory and implementation with computer science. Cyclic group theory deserves increased recognition. In addition, by providing definitions, evaluations, proofs, and concrete examples, this paper also endeavors to become a point of reference for readers to understand, consolidate, and further grasp cyclic group theory and its uses.

Almost elusive classical groups

Let G be a transitive permutation group acting on a finite set Ω with | Ω | ⩾ 2 . An element of G is said to be a derangement if it has no fixed points on Ω, and by a theorem of Jordan from 1872, G contains such an element. In particular, by a theorem of Fein, Kantor and Schacher, G contains a derangement of prime power order. Nevertheless there exist groups in which there are no derangements of prime order, these groups are called elusive groups. Defining a natural extension of this we say G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In recent work with Burness, we reduced the problem of determining the almost elusive quasiprimitive groups to the almost simple and 2-transitive affine cases. Additionally we classified the primitive almost elusive almost simple groups with socle an alternating group , a sporadic group or a group of Lie type with (twisted) Lie rank equal to 1. In this paper we complete the classification of the primitive almost elusive almost simple classical groups.

The Hall π-Subgroups of Some of the Classical Simple Groups

The Hall π-Subgroups of Some of the Classical Simple 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.

On flag‐transitive 2‐(k2,k,λ) $({k}^{2},k,\lambda )$ designs with λ∣k $\lambda | k$

AbstractIt is shown that, apart from the smallest Ree group, a flag‐transitive automorphism group of a 2‐ design , with , is either an affine group or an almost simple classical group. Moreover, when is the smallest Ree group, is isomorphic either to the 2‐ design or to one of the three 2‐ designs constructed in this paper. All the four 2‐designs have the 36 secants of a non‐degenerate conic of as a point set and 6‐sets of secants in a remarkable configuration as a block set.

On the orbital diameter of groups of diagonal type

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify all finite simple groups with covering number and conjugacy width at most 3. We also prove some general bounds on the covering number and conjugacy width of groups of Lie type.

On residuals of finite groups

AbstractTheorem C in [S. Dolfi, M. Herzog, G. Kaplan and A. Lev, The size of the solvable residual in finite groups,Groups Geom. Dyn.1(2007), 4, 401–407] asserts that, in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order and that the inequality is sharp. Inspired by this result and some of the arguments in the above article, we establish the following generalisation: if 𝔛 is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups andX¯\overline{\mathfrak{X}}is the extension-closure of 𝔛, then there exists an (explicitly known and optimal) constant 𝛾 depending only on 𝔛 such that, for all non-trivial finite groups 𝐺 with trivial 𝔛-radical,|GX¯|&gt;|G|γ\lvert G^{\overline{\mathfrak{X}}}\rvert&gt;\lvert G\rvert^{\gamma}, whereGX¯G^{\overline{\mathfrak{X}}}is theX¯\overline{\mathfrak{X}}-residual of 𝐺. WhenX=N\mathfrak{X}=\mathfrak{N}, the class of finite nilpotent groups, it follows thatX¯=S\overline{\mathfrak{X}}=\mathfrak{S}, the class of finite soluble groups; thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J. G. Thompson’s classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations 𝔛 of full characteristic such thatS⊂X¯⊂E\mathfrak{S}\subset\overline{\mathfrak{X}}\subset\mathfrak{E}, where 𝔈 denotes the class of all finite groups, thus providing applications of our main result beyond the reach of the above theorem.

On edge-primitive 3-arc-transitive graphs

This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.

Composition Factors of the Finite Groups Isospectral to Simple Classical Groups

Isospectral are the groups with coinciding sets of element orders. We prove that no finite group isospectral to a finite simple classical group has the exceptional groups of types $ E_{7} $ and $ E_{8} $ among its nonabelian composition factors.

Laws of the lattice of all $${\mathfrak {X}}$$-local formations of finite groups

Let $${\mathfrak {X}}$$ be a class of simple groups with a completeness property $$\pi ({\mathfrak {X}}) = \mathrm {char} \, {\mathfrak {X}}$$ . A formation is a class of finite groups closed under taking homomorphic images and finite subdirect products. Forster introduced the concept of $${\mathfrak {X}}$$ -local formation in order to obtain a common extension of well-known Gaschutz–Lubeseder–Schmid, and Baer theorems (Publ Mat Univ Autonoma Barcelona 29(2–3):39–76, 1985). In the present paper, it is shown that any law of the lattice of all formations is true in the lattice of all $${\mathfrak {X}}$$ -local formations.

Hitchin Systems on Hyperelliptic Curves

We describe a class of spectral curves and find explicit formulas for the Darboux coordinates of Hitchin systems corresponding to classical simple groups on hyperelliptic curves. We consider in detail the systems with rank $$2$$ groups on genus $$2$$ curves.

Arithmetic Subgroups and Applications

Arithmetic subgroups are an important source of discrete groups acting freely on manifolds. We need to know that there exist many torsion-free 푺푺L(ퟐퟐ,ℝ) is an “arithmetic” subgroup of 푺푺L(ퟐퟐ,ℝ). The other arithmetic subgroups are not as obvious, but they can be constructed by using quaternion algebras. Replacing the quaternion algebras with larger division algebras yields many arithmetic subgroups of 푺푺L(풏풏,ℝ), with 풏풏&gt;2. In fact, a calculation of group cohomology shows that the only other way to construct arithmetic subgroups of 푺푺L(풏풏, ℝ) is by using arithmetic groups. In this paper justifies Commensurable groups, and some definitions and examples,ℝ-forms of classical simple groups over ℂ, calculating the complexification of each classical group, Applications to manifolds. Let us start with 푺푺푺푺(푛푛,ℂ). This is already a complex Lie group, but we can think of it as a real Lie group of twice the dimension. As such, it has a complexification.

The lattice properties of 𝔛-local formations of finite groups

Let [Formula: see text] be a class of simple groups with a completeness property [Formula: see text]. Förster introduced the concept of [Formula: see text]-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer [Publ. Mat. UAB 29(2–3) (1985) 39–76]. In this paper, it is proved that the lattice of all [Formula: see text]-local formations of finite groups is modular.

Algebraic lattices of partially saturated formations of finite groups

Let $${\mathfrak {X}}$$ be a class of simple groups with a completeness property $$\pi ({\mathfrak {X}}) = \mathrm {char} \, \mathfrak X$$. Forster introduced the concept of $${\mathfrak {X}}$$-local formation in order to obtain a common extension of well-known theorems of Gaschutz–Lubeseder–Schmid and Baer (Publ Mat Univ Autonoma Barcelona, 29(2–3), 1985). In the present paper, it is proved that the lattice of all $${\mathfrak {X}}$$-local formations of finite groups is algebraic.

Orbits of maximal invariant subgroups and solvability of finite groups

Let A and G be finite groups having coprime orders and suppose that A acts on G via automorphisms. We give some solvability criteria for G according to the number of orbits that appear by the action of the fixed point subgroup CG(A) on the set of maximal A-invariant subgroups of G, and likewise, on the set of non-nilpotent maximal A-invariant subgroups. We also obtain some characterizations and further structure properties of these groups. In the course of our study we prove an independent result concerning maximal factorizations of classical simple groups.