Groups that are transitive on all partitions of a given shape

• Published in: Journal of algebraic combinatorics Vol. 42; no. 2; pp. 605 - 617
• Main Authors: Dobson, Edward, Malnič, Aleksander
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2015
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Unordered partition; Johnson graph; Transitive group; Cayley graph; Ordered partition
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-015-0593-2

Let [ n ] = K 1 ∪ ˙ K 2 ∪ ˙ ⋯ ∪ ˙ K r be a partition of [ n ] = { 1 , 2 , … , n } and set ℓ i = | K i | for 1 ≤ i ≤ r . Then the tuple P = { K 1 , K 2 , … , K r } is an unordered partition of [ n ] of shape [ ℓ 1 , … , ℓ r ] . Let P be the set of all partitions of [ n ] of shape [ ℓ 1 , … , ℓ r ] . Given a fixed shape [ ℓ 1 , … , ℓ r ] , we determine all subgroups G ≤ S n that are transitive on P in the following sense: Whenever P = { K 1 , … , K r } and P ′ = { K 1 ′ , … , K r ′ } are partitions of [ n ] of shape [ ℓ 1 , … , ℓ r ] , there exists g ∈ G such that g ( P ) = P ′ , that is, { g ( K 1 ) , … , g ( K r ) } = { K 1 ′ , … , K r ′ } . Moreover, for an ordered shape, we determine all subgroups of S n that are transitive on the set of all ordered partitions of the given shape. That is, with P and P ′ as above, g ( K i ) = K i ′ for 1 ≤ i ≤ r . As an application, we determine which Johnson graphs are Cayley graphs.