A new flag-transitive linear space
We construct a new flag-transitive $2$-$(496,4,1)$ design with automorphism group $\mathrm{PΓL}_2(32)$. This corrects an omission in the classification of the finite flag-transitive linear spaces.
math.CO↗
arXiv subjects
Seyed Hassan Alavi
Publications and source records attributed to Seyed Hassan Alavi.
On flag-transitive automorphism groups of $2$-designs with $λ$ prime
In this article, we study $2$-$(v,k,λ)$ designs $\mathcal{D}$ with $λ$ prime admitting flag-transitive and point-primitive almost simple automorphism groups $G$ with socle $T$ a finite exceptional simple group or a sporadic simple groups. If the socle of $G$ is a finite exceptional simple group, then we prove that $\mathcal{D}$ is isomorphic to one of two infinite families of $2$-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set $(v,b,r,k,λ)=(q^{2}+1,q^{2}(q^{2}+1)/(q-1),q^{2},q,q-1)$ design, where $q-1$ is a Mersenne prime, and the other is newly constructed in this paper and has parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ a Fermat prime. If $T$ is a sporadic simple group, then we show that $\mathcal{D}$ is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set $(v,b,r,k,λ)=(176,1100,50,2)$, $(12,22,11,6,5)$ or $(22,77,21,6,5)$.
math.GR↗
Almost simple groups as flag-transitive automorphism groups of 2-designs with λ = 2
In this article, we study $2$-designs with $λ=2$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type, and we prove that such a $2$-design does not exist. In conclusion, we present a classification of $2$-designs with $λ=2$ admitting flag-transitive and point-primitive automorphism groups of almost simple type, which states that such a $2$-design belongs to an infinite family of $2$-designs with parameter set $((3^n-1)/2,3,2)$ and $X=PSL_n(3)$ for some $n\geq 3$, or it is isomorphic to the $2$-design with parameter set $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(10,4,2)$, $(11,5,2)$, $(28,7,2)$, $(28,3,2)$, $(36,6,2)$, $(126,6,2)$ or $(176,8,2)$.
math.GR↗
Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs
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 $(v,k,λ)$ design with $λ$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,λ)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{AΓL}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $λ$ 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 $\mathrm{AΓL}_{1}(q)$.
math.GR↗
Higher-dimensional grid-imprimitive block-transitive designs
It was shown in 1989 by Delandtsheer and Doyen that, for a $2$-design with $v$ points and block size $k$, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if $v$ is small enough relative to $k$. Recently, exploiting a construction of block-transitive point-imprimitive $2$-designs given by Cameron and the last author, four of the authors studied $2$-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such $2$-designs to exist in terms of the parameters of the grid, and certain `array parameters' which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of $2$-designs for grids of dimensions three and four, and pose several open questions.
math.CO↗
Sporadic simple groups as flag-transitive automorphism groups of symmetric designs
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.
math.GR↗
On groups with the same character degrees as almost simple groups with socle small Ree groups
Let $G$ be a finite group and ${\rm cd}(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group with socle $H_{0}= \, ^{2}{\rm G}_{2}(q)$, where $q=3^{f}$ with $f\geq 3$ odd such that ${\rm cd}(G)={\rm cd}(H)$, then $G$ is non-solvable and the chief factor $G'/M$ of $G$ is isomorphic to $H_{0}$. If, in particular, $f$ is coprime to $3$, then $G'$ is isomorphic to $H_{0}$ and $G/{\bf Z}(G)$ is isomorphic to $H$.
math.GR↗
On quantitative structure of symplectic groups
The main aim of this article is to study the quantitative structure of projective symplectic groups $PSp_{4}(q)$ with $q>2$ even. Indeed, we prove that the groups $PSp_{4}(q)$ with $q>2$ even are uniquely determined by their orders and the set of the number of elements of the same order. This result links to the well-known J. G. Thompson's problem (1987) for finite simple groups.
math.GR↗
Block designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups
In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $AΓL_{1}(q)$.
math.GR↗
Block-transitive two-designs based on grids
We study point-block incidence structures $(\mathcal{P},\mathcal{B})$ for which the point set $\mathcal{P}$ is an $m\times n$ grid. Cameron and the fourth author showed that each block $B$ may be viewed as a subgraph of a complete bipartite graph $\mathbf{K}_{m,n}$ with bipartite parts (biparts) of sizes $m, n$. In the case where $\mathcal{B}$ consists of all the subgraphs isomorphic to $B$, under automorphisms of $\mathbf{K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for $(\mathcal{P},\mathcal{B})$ to be a $2$-design, and to be a $3$-design. We first re-interpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of $\mathbf{K}_{m,m}$. We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be $t$-designs, for $t=2, 3$, and give infinite families of examples illustrating that block-transitive, point-primitive $2$-designs based on grids exist for all values of $m$, and flag-transitive, point-primitive examples occur for all even $m$. This approach also allows us to construct a small number of block-transitive $3$-designs based on grids.
math.CO↗
Almost simple groups of Lie type and symmetric designs with $λ$ prime
In this article, we investigate symmetric $(v,k,λ)$ designs $\mathcal{D}$ with $λ$ 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 $(7,4,2)$, $(11,5,2)$, $(11,6,2)$ or $(45,12,3)$.
math.GR↗
Finite exceptional groups of Lie type and symmetric designs
In this article, we study symmetric $(v, k, λ)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle $X$ is a finite simple exceptional group of Lie type. We prove a reduction theorem, severely restricting the possible parameters of such designs. We also prove that the parameters $k$ and $λ$ are not coprime, and neither of these parameters can be prime. Moreover, if $λ$ is at most $100$, we show that there are two such parameters sets, namely, $(351,126,45)$ and $(378,117,36)$ for $G=X=G_{2}(3)$. 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.
math.GR↗
A classification of flag-transitive block designs
In this article, we investigate $2$-$(v,k,λ)$ designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,λ)$ design with $\gcd(k,λ)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq AΓL_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.
math.GR↗
A note on two families of $2$-designs arose from Suzuki-Tits ovoid
In this note, we give a precise construction of one of the families of $2$-designs arose from studying flag-transitive $2$-designs with parameters $(v,k,λ)$ whose replication numbers $r$ are coprime to $λ$. We show that for a given positive integer $q=2^{2n+1}\geq 8$, there exists a $2$-design with parameters $(q^{2}+1,q,q-1)$ and the replication number $q^{2}$ admitting the Suzuki group $\textsf{Sz(q)}$ as its automorphism group. We also construct a family of $2$-designs with parameters $(q^{2}+1,q(q-1),(q-1)(q^{2}-q-1))$ and the replication number $q^{2}(q-1)$ admitting the Suzuki groups $\textsf{Sz(q)}$ as their automorphism groups.
math.GR↗
Symmetries of biplanes
In this paper, we first study biplanes $\mathcal{D}$ with parameters $(v,k,2)$, where the block size $k\in\{13,16\}$. These are the smallest parameter values for which a classification is not available. We show that if $k=13$, then either $\mathcal{D}$ is the Aschbacher biplane or its dual, or $Aut(\mathcal{D})$ is a subgroup of the cyclic group of order $3$. In the case where $k=16$, we prove that $|Aut(\mathcal{D})|$ divides $2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13$. We also provide an example of a biplane with parameters $(16,6,2)$ with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.
math.GR↗
Flag-transitive block designs and finite exceptional simple groups of Lie type
In this article, we study $2$-designs with $\gcd(r,λ)=1$ 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.
math.GR↗
Flag-transitive block designs and unitary groups
In this article, we study $2$-designs with $\gcd(r, λ)=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D}, G)$, where $\mathcal{D}$ is a $2$-design with $\gcd(r, λ)=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n, q)$ with $(n, q)\neq (3, 2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design with parameters $(6, 3, 2)$, $(7, 3, 1)$, $(8, 4, 3)$, $(10, 6, 5)$, $(11, 5, 2)$ or $(28, 7, 2)$.
math.GR↗
Flag-transitive block designs with prime replication number and almost simple groups
In this article, we study $2$-designs with prime replication number admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine $2$-designs with prime replication number admitting an almost simple automorphism group.
math.GR↗