On the involution fixity of simple groups

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$ , which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$ . This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.

Flag-Transitive Non-Symmetric 2-Designs with $(r, \lambda)=1$ and Exceptional Groups of Lie Type

This paper determines all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G$ is the almost simple flag-transitive automorphism group of $\mathcal{D}$ with an exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of designs $\mathcal{D}$.

ON THE CHARACTERIZABILITY OF SOME FAMILIES OF FINITE GROUP OF LIE TYPE BY ORDERS AND VANISHING ELEMENT ORDERS

In this paper, we show that the following simple groups are uniquely determined by their orders and vanishing element orders: Ap-1(2), where p ̸= 3, 2Dp+1(2),where p ≥ 5, p ̸= 2m - 1, Ap(2), Cp(2), Dp(2), Dp+1(2) which for all of them p is anodd prime and 2p - 1 is a Mersenne prime. Also, 2Dn(2) where 2n-1 + 1 is a Fermatprime and n > 3, 2Dn(2) and Cn(2) where 2n + 1 is a Fermat prime. Then we give analmost general result to recognize the non-solvability of finite group H by an anologybetween orders and vanishing elemen orders of H and a finite simple group of Lie type.

On Splitting of the Normalizer of a Maximal Torus in E6(q)

Let G be a finite group of Lie type E6 over 𝔽q (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∊ W such that N(G, T )/T ≃ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.

ON THE SECOND-LARGEST SYLOW SUBGROUP OF A FINITE SIMPLE GROUP OF LIE TYPE

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except for an explicit list of exceptions and that $S$ is always ‘large’ in the sense that $|T|^{1/3}<|S|\leq |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that, for every $T$ and every prime divisor $r$ of $|T|$ with $r\neq p$, the order of the Sylow $r$-subgroup of $T$ is at most $|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$, where $\ell$ is the Lie rank of $T$.

On characters of Chevalley groups vanishing at the non-semisimple elements

Let [Formula: see text] be a finite simple group of Lie type. In this paper, we study characters of [Formula: see text] that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of [Formula: see text]. Such characters can be viewed as a natural generalization of the Steinberg character. For groups [Formula: see text] of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.

A characterisation of almost simple groups with socle 2E6(2) or M(22)

We show that the sporadic simple group M(22), the exceptional group of Lie type

The maximal subgroups of

Here we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type$E_{7}(2)$.Supplementary materials are available with this article.

Local identification of spherical buildings and finite simple groups of Lie type

We give a short proof of the uniqueness of finite spherical buildings of rank at least 3 in terms of the structure of the rank 2 residues and use this result to prove a result making it possible to identify an arbitrary finite group of Lie type from knowledge of its “parabolic structure” alone. Our proof also involves a connection between loops, “Latin chamber systems” and buildings.