In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,\lambda)$ design with $\lambda$ âŠ
In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,\lambda)$ design with $\lambda$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,\lambda)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $\lambda$ prime, which says that such an incidence structure is a projective space $\mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$.
Abstract Let be a nontrivial symmetric âdesign with , and let be a flagâtransitive automorphism group of . In this paper, we show that if is quasiprimitive on , then âŠ
Abstract Let be a nontrivial symmetric âdesign with , and let be a flagâtransitive automorphism group of . In this paper, we show that if is quasiprimitive on , then is of holomorph affine or almost simple type. Moreover, if is imprimitive on , then is of almost simple type. According to this observation and to the classification of the finite simple groups we determine all such symmetric designs and the corresponding automorphism groups. We conclude with two open problems and a conjecture.
Abstract In this article, we show that if is a nontrivial nonsymmetric design admitting a flagâtransitive pointâprimitive automorphism group G , then G must be an affine or almost simple âŠ
Abstract In this article, we show that if is a nontrivial nonsymmetric design admitting a flagâtransitive pointâprimitive automorphism group G , then G must be an affine or almost simple group. Moreover, if the socle of G is sporadic, then is the unique 2 â (176, 8, 2) design with , the HigmanâSims simple group.
In this article, we study symmetric designs with λ prime admitting a flag-transitive and point-primitive automorphism groups G of almost simple type. We prove that either D is one of âŠ
In this article, we study symmetric designs with λ prime admitting a flag-transitive and point-primitive automorphism groups G of almost simple type. We prove that either D is one of the six well-known examples of biplanes and triplanes, or D is the point- hyperplane design of PG(nâ1,q) with λ = (q^{nâ2}â1)/(qâ1) prime and X = PSLn(q).
Let [Formula: see text] be a subgroup of the full automorphism group of a [Formula: see text]-[Formula: see text] symmetric design [Formula: see text]. If [Formula: see text] is flag-transitive âŠ
Let [Formula: see text] be a subgroup of the full automorphism group of a [Formula: see text]-[Formula: see text] symmetric design [Formula: see text]. If [Formula: see text] is flag-transitive and point-primitive, then Soc[Formula: see text] cannot be [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].
In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting âŠ
In this article, we study symmetric designs admitting flag-transitive, point-imprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flag-transitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a point-imprimitive automorphism group and the remaining are point-primitive.
In this article, we study flag-transitive automorphism groups of non-trivial symmetric (v, k, λ) designs, where λ divides k and k ℠λ2. We show that such an automorphism group âŠ
In this article, we study flag-transitive automorphism groups of non-trivial symmetric (v, k, λ) designs, where λ divides k and k ℠λ2. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters v = λ2(λ + 2) and k = λ(λ + 1), for some positive integer λ. We also provide some examples in both possibilities.
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is âŠ
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive integer $\lambda$. We also provide some examples in both possibilities.
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is âŠ
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, \lambda)$ designs, where $\lambda$ divides $k$ and $k\geq \lambda^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=\lambda^{2}(\lambda+2)$ and $k=\lambda(\lambda+1)$, for some positive integer $\lambda$. We also provide some examples in both possibilities.
Abstract In this article, we study symmetric designs admitting flagâtransitive, pointâimprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs âŠ
Abstract In this article, we study symmetric designs admitting flagâtransitive, pointâimprimitive almost simple automorphism groups with socle sporadic simple groups. As a corollary, we present a classification of symmetric designs admitting flagâtransitive automorphism group whose socle is a sporadic simple group, and in conclusion, there are exactly seven such designs, one of which admits a pointâimprimitive automorphism group and the remaining are pointâprimitive.
Consider the direct product of symmetric groups $S_c\times S_n$ and its natural action on $\mathcal{P}=C\times N$, where $|C|=c$ and $|N|=n$. We characterize the structure of 2-designs with point set $\mathcal{P}$ âŠ
Consider the direct product of symmetric groups $S_c\times S_n$ and its natural action on $\mathcal{P}=C\times N$, where $|C|=c$ and $|N|=n$. We characterize the structure of 2-designs with point set $\mathcal{P}$ admitting flag-transitive, point-imprimitive automorphism groups $H\leq S_c\times S_n$. As an example of its applications, we show that $H$ cannot be any subgroup of $D_{2c}\times S_n$ or $S_c\times D_{2n}$. Besides, some families of 2-designs admitting flag-transitive automorphism groups $S_c\times S_n$ are constructed by using complete bipartite graphs and cycles. Two families of these also admit flag-transitive, point-primitive automorphism groups $S_c\wr S_2,$ a family of which attain the Cameron-Praeger upper bound $v=(k-2)^2$.
We study (v,k,λ)-symmetric designs having a flag-transitive, point-primitive automorphism group, with v = m2 and (k,λ) = t > 1, and prove that if D is such a design with âŠ
We study (v,k,λ)-symmetric designs having a flag-transitive, point-primitive automorphism group, with v = m2 and (k,λ) = t > 1, and prove that if D is such a design with m even admitting a flag-transitive, point-primitive automorphism group G, then either:D is a design with parameters ((2t+sâ1)2, (2t2 â (2âs)t)/s, (t2 â t)/s2)) with s â„ 1 odd, orG does not have a non-trivial product action.We observe that the parameters in (1), when s = 1, correspond to Menon designs.We also prove that if D is a (v,k,λ)-symmetric design with a flag-transitive, point-primitive automorphism group of product action type with v = ml and l â„ 2 then the complement of D does not admit a flag-transitive automorphism group.
Abstract It is shown that, if is a nontrivial 2â symmetric design, with , admitting a flagâtransitive automorphism group G of affine type, then , p an odd prime, and âŠ
Abstract It is shown that, if is a nontrivial 2â symmetric design, with , admitting a flagâtransitive automorphism group G of affine type, then , p an odd prime, and G is a pointâprimitive, blockâprimitive subgroup of . Moreover, acts flagâtransitively, pointâprimitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the âŠ
With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.
In this paper all integral solutions to the equation x 2 = 4q n -4q+1 when q is an odd prime are determined.This is done by working in a quadratic âŠ
In this paper all integral solutions to the equation x 2 = 4q n -4q+1 when q is an odd prime are determined.This is done by working in a quadratic field, using the unique factorization of ideals to reduce the problem to one about certain binary linear recurrences.One of the results is that the equation has no solutions with n > 2 if q > 5. 0. Introduction.In 1913 the Indian mathematician S. Ramanujan conjectured that the equation x 2 = 2 n -7 had only five solutions in positive integers.The solutions he gave were: Î = 3 4 5 7 15 x=l
Introduction. Bibliography. 282 pp. of tables. Appendix 1: Irrationalities and Conway polynomials. T. Breuer and S. Norton: Appendix 2: Improvements to the ATLAS of finite groups
Introduction. Bibliography. 282 pp. of tables. Appendix 1: Irrationalities and Conway polynomials. T. Breuer and S. Norton: Appendix 2: Improvements to the ATLAS of finite groups
Abstract We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) âŠ
Abstract We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines. Key Words: CameronâLiebler conjecture m-systemMatrix groupOvoidPrimitive prime divisorSpread2000 Mathematics Subject Classification: Primary 20G40Secondary 20C20, 20C33, 20C34 ACKNOWLEDGMENT We would like to thank Michael Giudici for many fruitful and stimulating conversations. This work forms part of an Australian Research Council Discovery Grant, for which the first author was supported. Notes Communicated by D. Easdown. Additional informationNotes on contributorsJohn Bamberg* *Current affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281-S22, 9000 Ghent, Belgium. Tim Penttila** **Current affiliation: Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA; E-mail: [email protected]
The permutation representations in the title are all determined, and no surprises are found to occur.
The permutation representations in the title are all determined, and no surprises are found to occur.
For each integer m , Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than n m , âŠ
For each integer m , Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than n m , provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.
This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections
This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections
Abstract This paper contains corrections to the tables of low-dimensional representations of quasi-simple groups published in the paper, âLow-dimensional representations of quasi-simple groupsâ, LMS Journal of Computation and Mathematics 4 âŠ
Abstract This paper contains corrections to the tables of low-dimensional representations of quasi-simple groups published in the paper, âLow-dimensional representations of quasi-simple groupsâ, LMS Journal of Computation and Mathematics 4 (2001) 22â63.