Exceptional Simple Groups Research Articles

On characterization by Gruenberg–Kegel graph of finite simple exceptional groups of Lie type

On characterization by Gruenberg–Kegel graph of finite simple exceptional groups of Lie type

On Medium-Rank Lie Primitive and Maximal Subgroups of Exceptional Groups of Lie Type

We study embeddings of groups of Lie type H H in characteristic p p into exceptional algebraic groups G \mathbf {G} of the same characteristic. We exclude the case where H H is of type P S L 2 \mathrm {PSL}_2 . A subgroup of G \mathbf {G} is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of G \mathbf {G} . With a few possible exceptions, we prove that there are no Lie primitive subgroups H H in G \mathbf {G} , with the conditions on H H and G \mathbf {G} given above. The exceptions are for H H one of P S L 3 ( 3 ) \mathrm {PSL}_3(3) , P S U 3 ( 3 ) \mathrm {PSU}_3(3) , P S L 3 ( 4 ) \mathrm {PSL}_3(4) , P S U 3 ( 4 ) \mathrm {PSU}_3(4) , P S U 3 ( 8 ) \mathrm {PSU}_3(8) , P S U 4 ( 2 ) \mathrm {PSU}_4(2) , P S p 4 ( 2 ) ′ \mathrm {PSp}_4(2)’ and 2 B 2 ( 8 ) {}^2\!B_2(8) , and G \mathbf {G} of type E 8 E_8 . No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of G \mathbf {G} . This has the consequence that, with the same exceptions, any almost simple group with socle H H , that is maximal inside an almost simple exceptional group of Lie type F 4 F_4 , E 6 E_6 , 2 E 6 {}^2\!E_6 , E 7 E_7 and E 8 E_8 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.

A Criterion for Nonsolvability of a Finite Group and Recognition of Direct Squares of Simple Groups

The spectrum ω(G) of a finite group G is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group G, there are four different primes such that ω(G) contains all their pairwise products but not a product of any three of these numbers, then G is nonsolvable. Using this result, we show that for q ⩾ 8 and q ≠ 32, the direct square Sz(q) × Sz(q) of the simple exceptional Suzuki group Sz(q) is uniquely characterized by its spectrum in the class of finite groups, while for Sz(32) × Sz(32), there are exactly four finite groups with the same spectrum.

Finite exceptional groups of Lie type and symmetric designs

In this article, we study symmetric designs D with parameters (v,k,λ) admitting a flag-transitive and point-primitive automorphism group G whose socle X is a finite simple exceptional group of Lie type. In conclusion, D is the symmetric design with parameters ((q6−1)/(q−1),q5,q4(q−1)) associated to the generalised hexagon H(q) or its dual and X=G2(q) with point-stabiliser ˆ[q5]:GL2(q), or D is the orthogonal design with parameters (351,126,45) or (378,117,36) respectively for ϵ=− or ϵ=+, and X=G2(3) with point-stabiliser SL3ϵ(3):2. Our analysis depends heavily on detailed information about actions of finite exceptional almost simple groups of Lie type on the cosets of their large maximal subgroups. In particular, properties derived in the paper about large subgroups and the subdegrees of such actions may be of independent interest.

Maximal 𝑃𝑆𝐿₂ Subgroups of Exceptional Groups of Lie Type

We study embeddings of P S L 2 ( p a ) \mathrm {PSL}_2(p^a) into exceptional groups G ( p b ) G(p^b) for G = F 4 , E 6 , 2 E 6 , E 7 G=F_4,E_6,{}^2\!E_6,E_7 , and p p a prime with a , b a,b positive integers. With a few possible exceptions, we prove that any almost simple group with socle P S L 2 ( p a ) \mathrm {PSL}_2(p^a) , that is maximal inside an almost simple exceptional group of Lie type F 4 F_4 , E 6 E_6 , 2 E 6 {}^2\!E_6 and E 7 E_7 , is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type A 1 A_1 inside the algebraic group. Together with a recent result of Burness and Testerman for p p the Coxeter number plus one, this proves that all maximal subgroups with socle P S L 2 ( p a ) \mathrm {PSL}_2(p^a) inside these finite almost simple groups are known, with three possible exceptions ( p a = 7 , 8 , 25 p^a=7,8,25 for E 7 E_7 ). In the three remaining cases we provide considerable information about a potential maximal subgroup.

Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

Double axes and subalgebras of Monster type in Matsuo algebras

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

Abstract A subgroup H of a group G is said to be pronormal in G if H and H g {H^{g}} are conjugate in 〈 H , H g 〉 {\langle H,H^{g}\rangle} for every g ∈ G {g\in G} . In this paper, we determine the finite simple groups of type E 6 ⁢ ( q ) {E_{6}(q)} and E 6 2 ⁢ ( q ) {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

On Thompson’s Conjecture for Finite Simple Exceptional Groups of Lie Type

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of exceptional Lie type or Tits group.

Flag-Transitive Block Designs and Finite Simple Exceptional Groups of Lie Type

In this article, we study 2-designs with $$\gcd (r,\lambda )=1$$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type. We obtain four infinite families of such designs and provide some examples in each of these families.

-TYPE SUBGROUPS CONTAINING REGULAR UNIPOTENT ELEMENTS

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X=\text{PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_{1}$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p)=(E_{6},13)$ or $(E_{7},19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the embedding of $X$ in $G$. We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.

On the involution fixity of exceptional groups of Lie type

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

Two examples of affine homogeneous varieties

The aim is to describe two homogeneous affine open sets in some projective spaces. The sets are well known, their groups of automorphisms contain simple exceptional groups of types \(E_6\) or \(E_7\), although the total groups of automorphisms are infinite-dimensional and both the sets are flexible. We prove existence of automorphisms not belonging to the connected components of unity, construct extended Weyl groups including these non-tame automorphisms. Our methods are based on classical combinatorics associated with 27 lines on non-singular cubic surfaces and with 56 exceptional curves on Del Pezzo surfaces of degree 2. Some traditional applications of Jordan algebras are used.

Symmetric invariants related to representations of exceptional simple groups

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an Abelian ideal, form a polynomial ring.

Spectra of Automorphic Extensions of Finite Simple Exceptional Groups of Lie Type

The spectrum ω (G) of a finite group G is the set of orders of elements of G. Let S be a simple exceptional group of type E6 or E7. We describe all finite groups G such that S ≤ G ≤ Aut S and ω (G) = ω (S) and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.

Almost Recognizability by Spectrum of Simple Exceptional Groups of Lie Type

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is isomorphic to a group $G$ squeezed between $L$ and its automorphism group, that is $L\leq G\leq \operatorname{Aut}L$; in particular, up-to isomorphism, there are only finitely many such groups. This assertion, together with a series of previously obtained results, implies that the same is true for every finite simple exceptional group except the group ${}^3D_4(2)$.

Unsolvable Block Transitive Automorphism Groups of 2-(v,11,1) Designs

The purpose of this paper is to discuss the classification of 2-(v,11,1) designs whose automorphism group G acts block transitive on its block. By using the result of the maximal sub-groups of the finite exceptional group of Lie type, we prove that the unsolvable block-transitive group G can not be isomorphic to exceptional simple group of Lie type.

Black box exceptional groups of Lie type II

If a black box group G is known to be isomorphic to an exceptional simple group Gˆ of Lie type of (twisted) rank >1, other than any F42(2e) or D43(2e), over a field of known size q, and if suitable SL2 and Discrete Log oracles are available when q is odd, then a polynomial-time Las Vegas algorithm is given that produces a constructive isomorphism between Gˆ and G.

Rational rigidity for

AbstractWe prove the existence of certain rationally rigid triples in${E}_{8}(p)$for good primes$p$(i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

On intersections of solvable Hall subgroups in finite simple exceptional groups of Lie type

It is proved that, if a Hall subgroup of an almost simple group whose socle is an exceptional group of Lie type is solvable, then there exist four groups conjugate to it whose intersection is trivial.