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Association schemes and permutation groups

Priscila P. Alejandro, R.A. Bailey, Peter J. Cameron
2003 Discrete Mathematics  
We examine these questions, and relate them to more familiar concepts of permutation group theory (such as generous transitivity) and association scheme theory (such as stratiÿability).  ...  Our main results are the determination of all regular groups having a unique minimal association scheme, and a classiÿcation of groups with no non-trivial association scheme.  ...  We do not know whether diagonal groups with n=2 are AS-friendly. The association scheme just constructed is not necessarily the minimal one.  ... 
doi:10.1016/s0012-365x(02)00798-7 fatcat:67wdd2yes5ecfgnv33nejdcxwe

Coherent configurations, association schemes and permutation groups [chapter]

Peter J. Cameron
2003 Groups, Combinatorics and Geometry  
The relationship between permutation groups and association schemes is quite subtle; there are groups which preserve no non-trivial association scheme, and other groups for which there is not a unique  ...  This paper gives a brief outline of the theory of coherent configurations and association schemes, and reports on some recent work on the connection between association schemes and permutation groups.  ...  Coherent configurations This section contains the definitions of coherent configurations and of various specialisations (including association schemes), and their connection with finite permutation groups  ... 
doi:10.1142/9789812564481_0004 fatcat:chafcop65rakfdtkhvk2ceoo5q

Computing normalizers of permutation groups efficiently using isomorphisms of association schemes

Izumi Miyamoto
2000 Proceedings of the 2000 international symposium on Symbolic and algebraic computation symbolic and algebraic computation - ISSAC '00  
This note presents an algorithm to speed up the computation of normalizers of permutation groups. It is an application of computation of isomorphisms of association schemes.  ...  For the definition and fundamental properties of association schemes, readers may refer to [1] . An association scheme used in this note is related to a transitive permutation group.  ...  We call N the group of isomorphisms of the system of association schemes. Lemma 1. Let K be a permutation group on Ω.  ... 
doi:10.1145/345542.345637 dblp:conf/issac/Miyamoto00 fatcat:jxiuiqfdk5cmvnyamde337cv3i

Computing Isomorphisms of Association Schemes and its Application

Izumi Miyamoto
2001 Journal of symbolic computation  
We also study an application of computing isomorphisms of association schemes to computing normalizers of permutation groups.  ...  Isomorphisms of association schemes are isomorphisms of edge-labeled regular graphs permitting to permute the labels.  ...  Then two association schemes A and B are isomorphic if B = P −1 A σ P for some permutation matrix P and some permutation σ on the set {1, 2, . . . , d} of relation numbers, where A σ is the matrix obtained  ... 
doi:10.1006/jsco.2001.0450 fatcat:ylck7gllxjaavjyvpwkcba2l2y

Association schemes for diagonal groups [article]

Peter J. Cameron, Sean Eberhard
2020 arXiv   pre-print
A transitive permutation group G is said to be AS-free if there is no non-trivial association scheme admitting G as a group of automorphisms.  ...  For any finite group G, and any positive integer n, we construct an association scheme which admits the diagonal group D_n(G) as a group of automorphisms.  ...  AS-free groups As noted above, a permutation group is AS-free if it preserves no non-trivial association scheme.  ... 
arXiv:1905.06569v2 fatcat:l72jqhgvrba73esxz3bbugingm

Classification of association schemes of small order

A. Hanaki, I. Miyamoto
2003 Discrete Mathematics  
It also classiÿes all the association schemes of order 24 -28 whether their automorphism groups are transitive or not, and whether they are group case or not.  ...  Some more properties of the obtained association schemes are computed.  ...  The automorphism group of A is the permutation group generated by the permutation matrices P satisfying A = P −1 AP and it is denoted by Aut(A).  ... 
doi:10.1016/s0012-365x(02)00551-4 fatcat:nz2opupxijg4jheyrcjfx553hm

Page 3825 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
Given a transitive permutation group G on a finite set X, the set of all 2-orbits of G on X (orbits of G on X x X) gives an association scheme, called a group-case association scheme.  ...  The paper under review provides an algorithm for computing iso- morphisms of association schemes. Then it applies the algorithm to compute some normalizers of certain permutation groups.  ... 

Page 6775 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
Group association schemes provide a natural link between finite group theory and finite scheme theory. Their basic relations are defined via the conjugacy classes of a group.  ...  (iii) Up to isomorphism, there are exactly three association schemes having the same intersection numbers as the group association scheme of the symmetric group Sym(4).  ... 

Construction of Association Scheme Using Some (123)-avoiding Class of Aunu Patterns

MS Magami, AA Ibrahim
2011 Nigerian Journal of Basic and Applied Sciences  
The finding in this research has shown that there is some interrelationship between the succession scheme used in generating Aunu numbers and the concept of association scheme.  ...  .: +234(0)81351008] 5 ABSTRACT: This paper presents some backgrounds research on association scheme using a class of (123)-avoiding pattern of Aunu numbers as an application area.  ...  in the compensation schemes there by forming a cyclic group from permutation of various succession models.  ... 
doi:10.4314/njbas.v19i1.69338 fatcat:ilfsrok4ofdwrohspfe6qh2aie

Computation of isomorphisms of coherent configurations

Izumi Miyamoto
2010 Ars Mathematica Contemporanea  
A transitive permutation group forms an association scheme while a permutation group which may not be transitive forms a coherent configuration.  ...  A program computing isomorphisms between association schemes was applied to speed up the computation of normalizers of permutation groups.  ...  As we mentioned in the introduction, a popular way to construct coherent configurations and association schemes is from permutation groups. Let G be a permutation group on Ω.  ... 
doi:10.26493/1855-3974.44.856 fatcat:bpzsangwzfar3brzzc7huboo64

Fission and fusion schemes and the action of the diagonal group D(T, n)

Priscila Alejandro Kazanidis
2008 Science Diliman  
The orbitals of a subgroup D(T, n) of the full automorphism group of the association schemes served as building blocks for the fusion schemes.  ...  This paper is a closer look at associate classes of an association scheme.  ...  Diagonal groups D(T, n) where n ≥ 8 and T abelian are generously transitive permutation groups. All orbitals in Orbl(Ω, D(T, n)) are symmetric and this gives a commutative association scheme.  ... 
doaj:2b8e82d3cc58479c8a40a14533700e1f fatcat:fxxzdoxjkvashil3dvyyu25dua


2006 Journal of the London Mathematical Society  
We also show how these ideas lead to a combinatorial basis for the Bose-Mesner algebra of the association scheme of the symmetric group and a design system attached to this association scheme.  ...  For example, if λ = (n − t, 1, . . . , 1), then a λ-transitive group is the same as a t-transitive permutation group, and if λ = (n − t, t), then we recover the t-homogeneous permutation groups.  ...  The authors wish to thank Peter Cameron, Jon Hall, Robert Liebler, Spyros Magliveras and Paul Terwilliger for helpful comments.  ... 
doi:10.1112/s0024610705022441 fatcat:tkdswyeyoje7hfe26cxpy6tsgq

Construction of association schemes and designs from finite groups

I.M Chakravarti, S Ikeda
1972 Journal of combinatorial theory. Series A  
A necessary and sufficient condition is given for the partition of a finite group to define an association scheme.  ...  Using this condition, several classes of association schemes and balanced incomplete designs have been constructed. *  ...  group divisible association scheme.  ... 
doi:10.1016/0097-3165(72)90027-1 fatcat:fqahu4zwkjbblibkd7o47i2uoa

Non-Associative Property of 123-Avoiding Class of Aunu Permutation Patterns

Aminu Alhaji Ibrahim, Sa'idu Isah Abubakar
2016 Advances in Pure Mathematics  
patterns of Aunu permutations using a pairing scheme.  ...  This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns.  ...  The 123-avoiding class of the Aunu permutation patterns which have been found to be of both combinatorics and group theoretic importance [1] can also be used to construct some structures which are non-associative  ... 
doi:10.4236/apm.2016.62006 fatcat:vwq3q7bvobdopgrtpnazwupiua

Page 1807 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
Kh.] (2-AOS-OK) Galois correspondence between permutation groups and cellular rings (association schemes). Graphs Combin. 6 (1990), no. 4, 303-332.  ...  Summary: “The notion of fusion in association schemes is de- veloped and then applied to group schemes. It is shown that the association schemes derived elsewhere [H. C. Agrawal and C. R.  ... 
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