Permutation groups, simple groups and sieve methods

We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, is ∼ hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n ≤ x of subgroups of abelian simple groups). We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are Sn and An in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982. Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes

Elusive Codes in Hamming Graphs

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case. We also pose several questions regarding elusive pairs

A Simplified Analysis of Radiant Heat Loss Through Projecting Fenestration Products

© 2001 ASHRAE (www.ashrae.org). Published in ASHRAE Transactions, Volume 107, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAE's prior written permissioCurrent window analysis algorithms can deal with many features, including low-e coatings and substitute fill gases. These methods were developed for products with planar glazings. Results can be generated for projecting products such as greenhouse windows, but the indoor-side heat transfer coefficient must be reduced to reflect differences in convection and radiant exchange for this geometry. Two simplified models are developed for radiant heat loss to projecting windows and are shown to agree well with a pseudo three-dimensional multi-element computer-based calculation. It is confirmed that the indoor-side heat transfer coefficient does not need to be accurately known to characterize a well-insulated window. More research is needed to quantify indoor-side convective heat loss before radiant exchange models can be verified and projecting products can be well characterized in general.Natural Sciences and Engineering Research Council of Canada || Natural Resources Canad

Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite 22-arc-transitive graph which is not a Cayley graph

A \emph{mixed dihedral group} is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper we give a sufficient condition such that the automorphism group of the Cayley graph \Cay(H,(X\cup Y)\setminus\{1\}) is equal to $H: A(H,X,Y)$, where $A(H,X,Y)$ is the setwise stabiliser in \Aut(H) of $X\cup Y$. We use this criterion to resolve a questions of Li, Ma and Pan from 2009, by constructing a $2$-arc transitive normal cover of order $2^{53}$ of the complete bipartite graph \K_{16,16} and prove that it is \emph{not} a Cayley graph.Comment: arXiv admin note: text overlap with arXiv:2303.00305, arXiv:2211.1680

Diagonal groups and arcs over groups

Partially supported by Simons Foundation Collaboration Grant 359872 and by Fundacaopara a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) grant PTDC/MAT-PUR/31174/2017. Australian Research Council Discovery Grant DP160102323.In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m≥2, a set of m+1 partitions of a set Ω, any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m=2), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m+r partitions with r≥2, any m of which are minimal elements of a Cartesian lattice. If m=2, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m>2, things are more restricted. Any m+1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m+r in the (m−1)-dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.Publisher PDFPeer reviewe

The geometry of diagonal groups

Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3).Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes, and is potentially of interest. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to the synchronization property of permutation groups.PostprintPeer reviewe