Three types of inclusions of innately transitive permutation groups into wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that corresponds to the product action of the wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also give a fairly general construction of inclusions for each type.     

The Bartle-Dunford-Schwartz theorem and projection operators in certain banach lattices

The projection map used successfully by Kakutani in his classical characterization of abstract L-spaces is used to establish a variant for certain Banach lattices of a theorem of Bartle, Dunford and Schwartz which dealth with absolute continuity in spaces of measures. Entrata in Redazione il 27 settembre 1999.     

Vocal Fold Mucosal Tears: Maintaining a High Clinical Index of Suspicion

Vocal fold mucosal tears have been discussed in the literature rarely, although they are not uncommon clinically. Disruptions in the epithelium usually follow trauma that may result from voice abuse and/or misuse, coughing, and other causes. A high index of suspicion is necessary to avoid missing vocal fold mucosal tears, and strobovideolaryngoscopy is indispensable in making the diagnosis. A brief period of complete voice rest is the standard of care and appears to be helpful in avoiding adverse sequelae and advancing the healing process, but there are no scientific studies to confirm its efficacy. Mucosal tears may heal completely or may be followed by the development of vocal fold masses, scar, and permanent dysphonia.     

An infinite family of biquasiprimitive 2-arc transitive cubic graphs

A new infinite family of bipartite cubic 3-arc transitive graphs is constructed and studied. They provide the first known examples admitting a 2-arc transitive vertex-biquasiprimitive group of automorphisms for which the index two subgroup fixing each half of the bipartition is not quasiprimitive on either bipartite half.     

Sporadic neighbour-transitive codes in Johnson graphs

We classify the neighbour-transitive codes in Johnson graphs $J(v,k)$ of minimum distance at least three which admit a neighbour-transitive group of automorphisms that is an almost simple two-transitive group of degree $v$ and does not occur in an infinite family of two-transitive groups. The result of this classification is a table of 22 codes with these properties. Many have relatively large minimum distance in comparison to their length $v$ and number of code words. We construct an additional five neighbour-transitive codes with minimum distance two admitting such a group. All 27 codes are $t$ -designs with $t$ at least two.     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

Two‐arc closed subsets of graphs

A subset of vertices of a graph is said to be 2‐arc closed if it contains every vertex that is adjacent to at least two vertices in the subset. In this paper, 2‐arc closed subsets generated by pairs of vertices at distance at most 2 are studied. Several questions are posed about the structure of such subsets and the relationships between two such subsets, and examples are given from the class of partition graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 350–364, 2003