Related papers: String C-group representations of transitive Group…

If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $\Gamma$ is at most $n/2+1$, with finitely many exceptions…

If $G$ is a transitive group of degree $n$ having a string C-group of rank $r\geq (n+3)/2$, then $G$ is necessarily the symmetric group $S_n$. We prove that if $n$ is large enough, up to isomorphism and duality, the number of string…

We prove that for any integer $n\geq 12$, and for every $r$ in the interval $[3, \ldots, \lfloor (n-1)/2\rfloor]$, the group $A_n$ has a string C-group representation of rank $r$ therefore showing that the only alternating group whose set…

We determine the ranks of string C-group representations of the groups ${\rm PSp}(4,\mathbb{F}_q)\cong\Omega(5,\mathbb{F}_q)$, and comment on those of higher-dimensional symplectic and orthogonal groups.

In this paper, string C-groups of all ranks $3 \leq r \leq \frac{n}{2}$ are provided for each alternating group $A_n$, $n \geq 12$. As the string C-group representations of $A_n$ have also been classified for $n \leq 11$, and it is known…

We prove that if G is a string C-group of rank 4 and G is isomorphic to L_2(q) with q a prime power, then q must be 11 or 19. The polytopes arising are Grunbaum's 11-cell of type {3,5,3} for L_2(11) and Coxeter's 57-cell of type {5,3,5} for…

We show that a rank reduction technique for string C-group representations first used for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on…

The study of string C-group representations of rank at least $n/2$ for the symmetric group $S_n$ has gained a lot of attention in the last fifteen years. In a recent paper, Cameron et al. gave a list of permutation representation graphs of…

This survey paper aims at giving the state of the art in the study of string C-group representations of almost simple groups. It also suggest a series of problems and conjectures to the interested reader.

We give a rank augmentation technique for rank 3 string C-group representations of the symmetric group $S_n$ and list the hypotheses under which it yields a valid string C-group representation of rank 4 thereof.

Up to isomorphism and duality, there are exactly two non-degenerate abstract regular polytopes of rank greater than $n-3$, one of rank $n-1$ and one of rank $n-2$, with automorphism groups that are transitive permutation groups of degree…

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among…

Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxy\cdots$ (of even or odd length) or a word of the form $yxyx\cdots$ (of even or…

We classify C-groups of ranks $n-1$ and $n-2$ for the symmetric group $S_n$. We also show that all these C-groups correspond to hypertopes, that is, thin, residually connected flag-transitive geometries. Therefore we generalise some similar…

Given a finite group $G$ and a conjugacy class of involutions $X$ of $G$, we define the commuting involution graph $\mathcal{C}(G,X)$ to be the graph with vertex set $X$ and $x,y \in X$ adjacent if and only if $x \neq y$ and $xy =yx$. In…

A code ${\mathcal C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and ${\mathcal C}$ is $2$-neighbour-transitive if the automorphism group $G={\rm Aut}({\mathcal C})$ acts transitively on each of the sets ${\mathcal C}$,…

In this paper, we present a classification of $2$-designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with…

We study the representations of the commutator subgroup of the braid group with n strands in the symmetric group of degree r. Motivated by some experimental results, we conjecture that for n>r, every such representation is trivial.

Let $G$ be a group and let $\rho\colon G\to\operatorname{Sym}(V)$ be a permutation representation of $G$ on a set $V$. We prove that there is a faithful $G$-coalgebra $C$ such that $G$ arises as the image of the restriction of…

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime…