Finite Simple Exceptional 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

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.

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.

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 ${}^3D_4(2)$.

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.

Recognizability of groups G 2(q) by spectrum

Two groups are said to be isospectral if they have equal sets of element orders. It is proved that for every finite simple exceptional group L = G 2(q) of Lie type, any finite group G isospectral to L must be isomorphic to L.

Triangle generation of finite exceptional groups of low rank

Given a triple ( p 1 , p 2 , p 3 ) of primes, the object of this paper is the study of the space Hom ( T p 1 , p 2 , p 3 , G ) of homomorphisms from the triangle group T p 1 , p 2 , p 3 to a finite simple exceptional group G of Lie type B 2 2 , G 2 2 , G 2 or D 4 3 . With a few exceptions, we give precise asymptotic estimates for the size of Hom ( T p 1 , p 2 , p 3 , G ) and determine the limiting probability that a randomly chosen homomorphism from T p 1 , p 2 , p 3 to G is surjective as | G | → ∞ .

(2,3)-Generation of Exceptional Groups

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(Z). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E7 and E8 over fields of odd characteristic occur as Galois groups of geometric extensions of Qab(t), where Qab denotes the maximal Abelian extension field of Q. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

Minimal permutation representations of finite simple exceptional groups of typesE 6,E 7, andE 8

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G2 and F4.

Generation of exceptional groups of Lie-type

In this paper we prove two theorems concerning the generation of a finite exceptional group of Lie-type GF. The first is: there is a semisimple element s such that for ‘nearly all’ elements x ∈ GFthe elements s and x generate the group GF. The second theorem we prove is: if G is a finite simple exceptional group of Lie-type not of type E6 or 2E6, then it is generated by three involutions.