Suborbits in Transitive Permutation Groups

With any graph we can associate a group, namely its automorphism group; this acts naturally as a permutation group on the vertices of the graph. The converse idea, that of reconstructing a graph (or a family of graphs) from a transitive permutation group, has been developed by C.C. Sims, D.G. Higman, and many other people, and is the subject of the present survey. In his lecture notes [23], Higman has axiomatised the combinatorial objects that arise from permutation groups in this way, under the name coherent configurations-, but I shall discuss only the case where a group is present. My own introduction to the theory was via the unpublished paper of P.M. Neumann [30].

[1]  Noboru Ito.  On uniprimitive permutation groups of degree 2p , 1967 .

[2]  D. G. Higman.  Primitive rank 3 groups with a prime subdegree , 1965 .

[3]  Peter J. Cameron,et al.  Bounding the rank of certain permutation groups , 1972 .

[4]  H. Wielandt,et al.  Permutation groups through invariant relations and invariant functions , 1969 .

[5]  Walter Feit,et al.  Characters of finite groups , 1965 .

[6]  Peter J. Cameron.  Extending Symmetric Designs , 1973, J. Comb. Theory, Ser. A.

[7]  E. Bannai,et al.  On finite Moore graphs , 1973 .

[8]  W. A. Manning.  On Simply Transitive Primitive Groups. , 1927, Proceedings of the National Academy of Sciences of the United States of America.

[9]  F. C. Piper,et al.  FINITE GROUPS OF AUTOMORPHISMS , 1974 .

[10]  Walter Feit,et al.  The nonexistence of certain generalized polygons , 1964 .

[11]  D. G. Higman.  Intersection matrices for finite permutation groups , 1967 .

[12]  Peter J. Cameron.  Permutation Groups with Multiply Transitive Suborbits , 1972 .

[13]  W. Kantor,et al.  Finite groups with a split BN-pair of rank 1. II , 1972 .

[14]  Wolfgang Knapp,et al.  On the point stabilizer in a primitive permutation group , 1973 .

[15]  W. Burnside,et al.  On some Properties of Groups of Odd Order , 1900 .

[16]  Surinder K. Sehgal,et al.  Subnormal subgroups and permutation groups : lectures given at the Ohio State University, Columbus, Ohio, in the Spring of 1971 , 1971 .

[17]  Norman Biggs,et al.  On Trivalent Graphs , 1971 .

[18]  Anthony Gardiner.  Doubly primitive vertex stabilisers in graphs , 1974 .

[19]  Dale M. Mesner,et al.  A New Family of Partially Balanced Incomplete Block Designs with Some Latin Square Design Properties , 1967 .

[20]  Derek H. Smith.  Bounding the diameter of a distance-transitive graph , 1974 .

[21]  H. Wielandt,et al.  Finite Permutation Groups , 1964 .

[22]  Helmut Bender,et al.  Transitive gruppen gerader ordnung, in denen jede involution genau einen punkt festläβt , 1971 .

[23]  W. J. Wong,et al.  Determination of a class of primitive permutation groups , 1967 .

[24]  Helmut Wielandt.  Über den Transitivitätsgrad von Permutationsgruppen , 1960 .

[25]  Jacques Tits,et al.  Sur la trialité et certains groupes qui s’en déduisent , 1959 .

[26]  W. T. Tutte.  A family of cubical graphs , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  Permutation Groups with Multiply‐Transitive Suborbits, II , 1974 .

[28]  John G. Thompson,et al.  Bounds for orders of maximal subgroups , 1970 .

[29]  R. M. Damerell.  On Moore graphs , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  Charles C. Sims,et al.  Graphs and finite permutation groups , 1967 .

[31]  A. Gardiner.  ARC TRANSITIVITY IN GRAPHS III , 1974 .

[32]  A. Lempel.  m-ary closed sequences , 1971 .

[33]  Derek H. Smith.  Primitive and imprimitive graphs , 1971 .

[34]  D. Djoković,et al.  On regular graphs, VI , 1973 .

[35]  Primitive groups with most suborbits doubly transitive , 1973 .

[36]  William M. Kantor,et al.  2-Transitive Designs , 1975 .

[37]  A. Gardiner,et al.  ARC TRANSITIVITY IN GRAPHS , 1973 .

[38]  G. Higman,et al.  Embedding Theorems for Groups , 1949 .

[39]  W. A. Manning.  A theorem concerning simply transitive primitive groups , 1929 .

[40]  W. T. Tutte.  On the Symmetry of Cubic Graphs , 1959, Canadian Journal of Mathematics.

[41]  William L. Quirin.  Primitive permutation groups with small orbitals , 1971 .

[42]  C. Sims.  Computational methods in the study of permutation groups , 1970 .

[43]  M. S. Smith,et al.  On rank 3 permutation groups , 1975 .

[44]  D. G. Higman.  Finite permutation groups of rank 3 , 1964 .

[45]  P. Delsarte,et al.  The Association Schemes of Coding Theory , 1975 .

[46]  Extension of some results of Manning and Wielandt on primitive permutation groups , 1971 .

[47]  J. L. Britton,et al.  THE WORD PROBLEM , 1963 .

[48]  D. G. Higman.  Combinatorial considerations about permutation groups , 1972 .

[49]  Primitive Permutationsgruppen vom Grad 2p , 1955 .

[50]  Mike D. Atkinson.  DOUBLY TRANSITIVE BUT NOT DOUBLY PRIMITIVE PERMUTATION GROUPS II , 1974 .