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Group actions: combinatorics, geometry and computation

Title
Group actions: combinatorics, geometry and computation
Funding
ARC | Federation Fellowships
Contract (GA) number
FF0776186
Start Date
2007/01/01
End Date
2012/12/31
Open Access mandate
no
Organizations
-
More information
http://purl.org/au-research/grants/arc/FF0776186

 

  • Pairwise transitive 2-designs

    Devillers, Alice; Praeger, Cheryl E. (2014)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims d...

    Line graphs and $2$-geodesic transitivity

    Devillers, Alice; Jin, Wei; Li, Cai Heng; Praeger, Cheryl E. (2012)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    For a graph $\Gamma$, a positive integer $s$ and a subgroup $G\leq \Aut(\Gamma)$, we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs ...

    A new solvability criterion for finite groups

    Dolfi, Silvio; Herzog, Marcel; Praeger, Cheryl E. (2010)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the ...

    Abelian coverings of finite general linear groups and an application to their non-commuting graph

    Azad, A.; Iranmanesh, M. A.; Praeger, C. E.; Spiga, P. (2010)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    In this paper we introduce and study a family $\mathcal{A}_n(q)$ of abelian subgroups of $\GL_n(q)$ covering every element of $\GL_n(q)$. We show that $\mathcal{A}_n(q)$ contains all the centralisers of cyclic matrices and equality holds if $q>n$. Also, for $q>2$, we prove a simple closed formula for the size of $\mathcal{A}_n(q)$ and give an upper bound if $q=2$. A subset $X$ of a finite group $G$ is said to be pairwise non-commuting if $xy\not=yx$, for distinct elements $x, y$ in $X$. As an...

    On graph-restrictive permutation groups

    Potocnik, Primoz; Spiga, Pablo; Verret, Gabriel (2011)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    Let $\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $\Gamma$ and let $L=G_v^{\Gamma(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. Then $(\Gamma,G)$ is said to be \emph{locally-$L$}. A transitive permutation group $L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the inequality $|G_{uv}|\leq c(L)$ h...

    Construction of long root SL(2,q)-subgroups in black box groups

    We present a one sided Monte--Carlo algorithm which constructs a long root $\sl_2(q)$-subgroup in $X/O_p(X)$, where $X$ is a black-box group and $X/O_p(X)$ is a finite simple group of Lie type defined over a field of odd order $q=p^k > 3$ for some $k\geqslant 1$. Our algorithm is based on the analysis of the structure of centralizers of involutions and can be viewed as a computational version of Aschbacher's Classical Involution Theorem. We also present an algorithm which determines whether t...

    Compositions of n Satisfying Some Coprimality Conditions

    Bubboloni, Daniela; Luca, Florian; Spiga, Pablo (2012)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    A k-composition of n is a sequence of length k of positive integers summing up to n. In this paper, we investigate the number of k-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic formula for the number of k-compositions having the first summand coprime to the others. Then, we estimate the number of k-compositions whose summands are all pairwise coprime.

    Neighbour transitivity on codes in Hamming graphs

    Gillespie, Neil I.; Praeger, Cheryl E. (2011)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. We call these codes \emph{neighbour transitive}. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we const...

    Entry-Faithful $2$-Neighbour Transitive Codes

    Gillespie, Neil I.; Giudici, Michael; Hawtin, Daniel R.; Praeger, Cheryl E. (2014)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    We consider a code to be a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of vertices, not in the code, at distance $s$ from some codeword, but not distance less than $s$ from any codeword. A $2$-neighbour transitive code is a code which admits a group $X$ of automorphisms which is transitive on the $s$-neighbours, for $s=1,2$, and transitive on the code itself. We give a classification of $2$-neighbour transitive codes, with minimum distance $\del...

    Diagonally Neighbour Transitive Codes and Frequency Permutation Arrays

    Gillespie, Neil I.; Praeger, Cheryl E. (2012)
    Projects: ARC | Group actions: combinatorics, geometry and computation (FF0776186)
    Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication, while at the same time maintaining a constant power output. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency per...
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