Generalized Commuting Groups

Abstract

In Generalized Musical Intervals and Transformations (1987), David Lewin describes the structure of commuting groups for transformation groups that have simply transitive actions. We extend Lewin's notion to transformation groups that have any type of action, including merely transitive, semiregular, faithful, and diagonal cases, among others. Three general situations arise: First, the commuting group for a group with an action that is merely transitive is isomorphic to the normalizer of a point stabilizer modulo the point stabilizer. Second, the commuting group for a diagonal subgroup of a direct product of orbit restrictions is a wreath product of the commuting group of one orbit restriction (as above) by the symmetric group on the set of orbits. Third, the commuting group for any group with any type of action is a direct product of such wreath products for each union of orbit restrictions that is a maximally embedded diagonal subgroup. We apply these concepts to commuting groups for familiar transformation groups, including the transposition group's and the transposition and inversion group's actions on assorted set-classes and unions of set-classes with varying symmetrical properties.