Related papers: Groups Obtained from $2-(n,4,3)$ Supersimple Desig…

To each supersimple $2-(n,4,\lambda)$ design $\mathcal{D}$ one associates a `Conway groupoid,' which may be thought of as a natural generalisation of Conway's Mathieu groupoid associated to $M_{13}$ which is constructed from $\mathbb{P}_3$.…

A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The…

To a set $\mathcal{B}$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's…

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined…

We construct a complex $\mathcal{L}_\bullet^\lambda$ resolving the irreducible representations $\mathcal{S}^{\lambda[n]}$ of the symmetric groups $S_n$ by representations restricted from $GL_n(k)$. This construction lifts to…

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log…

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,\Lambda)$ which are normalized by the elementary subgroup $EU_{2n}(R,\Lambda)$, under the condition that $R$ is…

Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved…

An original non-standard approach to describing the structure of a column stabilizer in a group of $n \times n$ matrices over a polynomial ring or a Laurent polynomial ring of $n$ variables is presented. The stabilizer is described as an…

We establish the existence of simple designs with parameters $2$-$(55,10,4)$, $3$-$(20,5,4)$, $3$-$(21,7,30)$, $4$-$(15,5,2)$, $4$-$(16,8,45)$, $5$-$(16,7,10)$, and $5$-$(17,8,40)$, which have previously been unknown. For the corresponding…

This paper investigates simple $3$-$(2^n+1,13,\lambda)$ designs admitting $\mathrm{PSL}$$(2,2^n)$ as an automorphism group. We determine all possible values of $\lambda$ by systematically analyzing the orbits of $13$-element subsets under…

In 1997, John Conway constructed a $6$-fold transitive subset $M_{13}$ of permutations on a set of size $13$ for which the subset fixing any given point was isomorphic to the Mathieu group $M_{12}$. The construction was via a…

In this article we give an self contained existence proof for J. Conway's sporadic simple group Co_1 [4] using the second author's algorithm [14] constructing finite simple groups from irreducible subgroups of GL_n(2). Here n = 11 and the…

Non-trivial $2$-$(k^{2},k,\lambda )$ designs, with $\lambda \mid k$, admitting a flag-transitive almost simple automorphism group are classified.

We study closed, connected, spin 4-manifolds up to stabilisation by connected sums with copies of $S^2 \times S^2$. For a fixed fundamental group, there are primary, secondary and tertiary obstructions, which together with the signature…

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…

We show the simplicity of the automorphism groups of the generic $n$-hypertournament and the semigeneric tournament, and determine the normal subgroups of the automorphism groups of several other ultrahomogeneous oriented graphs. We also…

This paper studies the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, it generalizes earlier work of Ribet and Moonen-Zarhin to completely determine the…

The classification of the $2$-designs with $\lambda=2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in \cite{ABDT}. The result is achieved by using conics and hyperovals…

With any integral lattice \Lambda in n-dimensional euclidean space we associate an elementary abelian 2-group I(\lambda) whose elements represent parts of the dual lattice that are similar to \Lambda. There are corresponding involutions on…