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

Seyed Hassan Alavi; Mohsen Bayat; Ashraf Daneshkhah

doi:10.37236/9366

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

The Electronic Journal of Combinatorics ◽

10.37236/9366 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Seyed Hassan Alavi ◽

Mohsen Bayat ◽

Ashraf Daneshkhah

Keyword(s):

Projective Space ◽

Simple Group ◽

Finite Simple Group ◽

Automorphism Groups ◽

Simple Groups ◽

Almost Simple Group ◽

Group Of Lie Type ◽

Symmetric Designs ◽

Almost Simple Groups ◽

Groups Of Lie Type

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 $(45,12,3)$.

On the shortest identity in finite simple groups of Lie type

Journal of Group Theory ◽

10.1515/jgt.2010.039 ◽

2011 ◽

Vol 14 (1) ◽

Cited By ~ 4

Author(s):

Uzy Hadad

Keyword(s):

Simple Group ◽

Finite Simple Group ◽

Simple Groups ◽

Finite Simple Groups ◽

Group Of Lie Type ◽

Groups Of Lie Type

AbstractWe prove that the length of the shortest identity in a finite simple group of Lie type of rank

Non-commuting graphs of certain almost simple groups

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500815 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950081

Author(s):

M. Jahandideh ◽

R. Modabernia ◽

S. Shokrolahi

Keyword(s):

Finite Group ◽

Abelian Group ◽

Simple Group ◽

Simple Groups ◽

Almost Simple Group ◽

Almost Simple Groups ◽

Commuting Graph ◽

Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

ON A CONJECTURE ON THE PERMUTATION CHARACTERS OF FINITE PRIMITIVE GROUPS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719001060 ◽

2019 ◽

Vol 102 (1) ◽

pp. 77-90

Author(s):

PABLO SPIGA

Keyword(s):

Fixed Point ◽

Finite Group ◽

Simple Group ◽

Simple Groups ◽

Alternating Group ◽

Permutation Groups ◽

Sporadic Simple Group ◽

Almost Simple Groups ◽

A Finite Group ◽

Groups Of Lie Type

Let $G$ be a finite group with two primitive permutation representations on the sets $\unicode[STIX]{x1D6FA}_{1}$ and $\unicode[STIX]{x1D6FA}_{2}$ and let $\unicode[STIX]{x1D70B}_{1}$ and $\unicode[STIX]{x1D70B}_{2}$ be the corresponding permutation characters. We consider the case in which the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{1}$ coincides with the set of fixed-point-free elements of $G$ on $\unicode[STIX]{x1D6FA}_{2}$, that is, for every $g\in G$, $\unicode[STIX]{x1D70B}_{1}(g)=0$ if and only if $\unicode[STIX]{x1D70B}_{2}(g)=0$. We have conjectured in Spiga [‘Permutation characters and fixed-point-free elements in permutation groups’, J. Algebra299(1) (2006), 1–7] that under this hypothesis either $\unicode[STIX]{x1D70B}_{1}=\unicode[STIX]{x1D70B}_{2}$ or one of $\unicode[STIX]{x1D70B}_{1}-\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{2}-\unicode[STIX]{x1D70B}_{1}$ is a genuine character. In this paper we give evidence towards the veracity of this conjecture when the socle of $G$ is a sporadic simple group or an alternating group. In particular, the conjecture is reduced to the case of almost simple groups of Lie type.

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

Forum Mathematicum ◽

10.1515/forum-2013-0055 ◽

2015 ◽

Vol 27 (5) ◽

Author(s):

Chris Parker ◽

M. Reza Salarian ◽

Gernot Stroth

Keyword(s):

Simple Group ◽

Simple Groups ◽

Exceptional Group ◽

Sporadic Simple Group ◽

Group Of Lie Type ◽

Almost Simple Groups

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

Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

Journal of Group Theory ◽

10.1515/jgth-2015-0051 ◽

2016 ◽

Vol 19 (2) ◽

Cited By ~ 1

Author(s):

Martino Garonzi ◽

Dan Levy ◽

Attila Maróti ◽

Iulian I. Simion

Keyword(s):

Simple Group ◽

Finite Simple Group ◽

Minimal Length ◽

Simple Groups ◽

Finite Simple Groups ◽

Sylow Subgroups ◽

Groups Of Lie Type

AbstractWe prove that every finite simple group

Almost simple groups with socle PSp4(q) as flag-transitive automorphism groups of symmetric designs

Discrete Mathematics ◽

10.1016/j.disc.2021.112615 ◽

2021 ◽

Vol 344 (12) ◽

pp. 112615

Author(s):

Seyed Hassan Alavi ◽

Mohsen Bayat ◽

Ashraf Daneshkhah ◽

Narges Okhovat

Keyword(s):

Automorphism Groups ◽

Simple Groups ◽

Symmetric Designs ◽

Almost Simple Groups ◽

Transitive Automorphism

GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

Forum of Mathematics Sigma ◽

10.1017/fms.2019.43 ◽

2020 ◽

Vol 8 ◽

Author(s):

ANDREA LUCCHINI ◽

CLAUDE MARION ◽

GARETH TRACEY

Keyword(s):

Finite Group ◽

Maximal Subgroup ◽

Simple Group ◽

Finite Groups ◽

Upper Bounds ◽

Maximal Subgroups ◽

Simple Groups ◽

Almost Simple Group ◽

Almost Simple Groups ◽

A Finite Group

For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

RECOGNITION OF THE PROJECTIVE GENERAL LINEAR GROUP PGL(2, q) BY ITS NONCOMMUTING GRAPH

Journal of Algebra and Its Applications ◽

10.1142/s0219498811004513 ◽

2011 ◽

Vol 10 (02) ◽

pp. 201-218 ◽

Cited By ~ 3

Author(s):

LIANGCAI ZHANG ◽

WUJIE SHI

Keyword(s):

Simple Group ◽

Linear Group ◽

General Linear Group ◽

Simple Groups ◽

General Linear ◽

Almost Simple Group ◽

Almost Simple Groups ◽

Vertex Set ◽

Noncommuting Graph ◽

Projective General Linear Group

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL (2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

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

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1903573p ◽

2019 ◽

pp. 573

Author(s):

Majedeh Pasdar ◽

Ali Iranmanesh

Keyword(s):

Finite Group ◽

Simple Group ◽

General Result ◽

Finite Simple Group ◽

Simple Groups ◽

Element Orders ◽

Group Of Lie Type ◽

Mersenne Prime

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.

GROUPS WITH THE SAME CHARACTER DEGREES AS SPORADIC ALMOST SIMPLE GROUPS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000253 ◽

2016 ◽

Vol 94 (2) ◽

pp. 254-265

Author(s):

SEYED HASSAN ALAVI ◽

ASHRAF DANESHKHAH ◽

ALI JAFARI

Keyword(s):

Finite Group ◽

Simple Group ◽

Irreducible Character ◽

Abelian Subgroup ◽

Character Degrees ◽

Simple Groups ◽

Sporadic Simple Group ◽

Almost Simple Group ◽

Almost Simple Groups ◽

A Finite Group

Let$G$be a finite group and$\mathsf{cd}(G)$denote the set of complex irreducible character degrees of$G$. We prove that if$G$is a finite group and$H$is an almost simple group whose socle is a sporadic simple group$H_{0}$and such that$\mathsf{cd}(G)=\mathsf{cd}(H)$, then$G^{\prime }\cong H_{0}$and there exists an abelian subgroup$A$of$G$such that$G/A$is isomorphic to$H$. In view of Huppert’s conjecture, we also provide some examples to show that$G$is not necessarily a direct product of$A$and$H$, so that we cannot extend the conjecture to almost simple groups.