On Hering Decomposition ofDKnInduced by Group Actions on Conjugacy Classes

A Hering decomposition of type t and order n is a factorization of the complete directed graph DKninto arc disjoint t -circuits for which the following conditions hold: (1) The vertices and arcs of DKnare partitioned into arc disjoint parallel classes, each containing n − 1 t circuits and a single vertex. (2) Any two parallel classes have exactly one edge (undirected arc) in common. For each parallel class Pi, 1 ≤ i ≤ n there corresponds a permutationσi acting on the letter set {1, 2,⋯ , n } as follows:σi (k) = l iff (k,l ) is an arc of Pi. Let G be the group generated byσ1 , σ2,⋯ , σn. The given configuration is inner transitive if G permutes the parallel classes transitively. In this paper we study the interaction between an inner-transitive Hering configuration and the structure of the corresponding group G. In particular, we classify all the pairs (n, t) for which an inner-transitive Hering configuration of type t and order n exists.