Related papers: Conway's groupoid and its relatives

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…

We study a construction of the Mathieu group $M_{12}$ using a game reminiscent of Loyd's ``15-puzzle''. The elements of $M_{12}$ are realized as permutations on~12 of the~13 points of the finite projective plane of order~3. There is 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…

It is shown that the supersymmetry-preserving automorphisms of any non-linear sigma-model on K3 generate a subgroup of the Conway group Co_1. This is the stringy generalisation of the classical theorem, due to Mukai and Kondo, showing that…

We contribute towards the classification programme for Conway groupoids associated to a $2-(n,4,\lambda)$ design. Our main results improve the known bounds for a hole stabilizer to be primitive, or to contain the alternating group, ${\rm…

In this paper we construct distance-regular graphs admitting a transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$ and $M_{24}$. From the code spanned by…

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…

In the spirit of Conway we define a groupoid starting from projective planes of order $q$, where $q$ is odd. The associated group of these groupoids is then investigated.

We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the…

John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…

In this paper we construct structures from Mathieu group $M_{11}$. We classify transitive $t$-designs with 11, 12 and 22 points admitting a transitive action of Mathieu group $M_{11}$. Thereby we proved the existence of designs with…

Let $\mathbb{M}$ be the monster group which is the largest sporadic finite simple group, and has first been constructed in 1982 by Griess. In 1985, Conway has constructed a 196884-dimensional representation $\rho$ of $\mathbb{M}$ with…

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts…

We address a natural question in noncommutative geometry, namely the rigidity observed in many examples, whereby noncommutative spaces (or equivalently their coordinate algebras) have very few automorphisms by comparison with their…

Consider the diagonal action of the projective group $\PGL_3$ on $n$ copies of ${\mathbb P}^2$. In addition, consider the action of the symmetric group $\Sigma_n$ by permuting the copies. In this paper we find a set of generators for the…

The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite…

The theory of composite mixtures consisting of $n$ constituents is framed within the schema provided by the notion of $n$-groupoid. The point of departure is the analysis of $n$-dimensional hypercubes and their skeletons, to each of whose…

This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. We encode geometric structures induced by transitive Lie…

The famous concyclicity theorem stated by John H. Conway is here reconsidered by means of a parametrisation of the associated triangular configuration with arbitrary triplets of real numbers ($\alpha$;$\beta$;$\gamma$). This theorem, thus…