arXiv subjects
Publications and source records attributed to Seyed Hassan Alavi.
In this article, we study symmetric $(v, k, λ)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PSU_{4}(q)$. We prove that there exist eight non-isomorphic such designs for which $λ\in\{3,6,18\}$ and $G$ is either $PSU_{4}(2)$, or $PSU_{4}(2):2$.
In this article, we study flag-transitive automorphism groups of non-trivial symmetric $(v, k, λ)$ designs, where $λ$ divides $k$ and $k\geq λ^2$. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters $v=λ^{2}(λ+2)$ and $k=λ(λ+1)$, for some positive integer $λ$. We also provide some examples in both possibilities.
For a finite group $G$ and a positive integer $n$, let $G(n)$ be the set of all elements in $G$ such that $x^{n}=1$. The groups $G$ and $H$ are said to be of the same (order) type if $G(n)=H(n)$, for all $n$. The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$, where $q=2^{2m+1}\geq 8$, then $G$ is isomorphic to $Sz(q)$. This addresses the well-known J. G. Thompson's problem (1987) for simple groups.
The main aim of this article is to study quantitative structure of small Ree Groups $^{2}G_{2}(q)$. Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order. As a consequence, we give a positive answer to J. G. Thompson's problem for simple groups $^{2}G_{2}(q)$.
Let G be a finite group and 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 whose socle is a sporadic simple group H0 such that cd(G) = cd(H), then G' = H0 and there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert's conjecture (2000), we also provide some examples to show that G is not necessarily a direct product of A and H, and hence we cannot extend this conjecture to almost simple groups.
Let $G$ be a finite group and $cd(G)$ 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 whose socle is Mathieu group such that $cd(G) =cd(H)$, then there exists an Abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. This study is heading towards the study of an extension of Huppert's conjecture (2000) for almost simple groups.
Triple factorisations of finite groups $G$ of the form $G=PQP$ are essential in the study of Lie theory as well as in geometry. Geometrically, each triple factorisation $G=PQP$ corresponds to a $G$-flag transitive point/line geometry such that `each pair of points is incident with at least one line'. We call such a geometry \emph{collinearly complete}, and duality (interchanging the roles of points and lines) gives rise to the notion of \emph{concurrently complete} geometries. In this paper, we study triple factorisations of the general linear group $\mathrm{GL}(V)$ as $PQP$ where the subgroups $P$ and $Q$ either fix a subspace or fix a decomposition of $V$ as $V_1\oplus V_2$ with $\dim(V_{1})=\dim(V_{2})$.