Maximal subgroups of the semigroup of partial symmetries of a regular polygon

Abstract

The semigroup of partial symmetries of a polygon $P$ is the collection of all distance-preserving bijections between subpolygons of $P$, with composition as the operation. Around every idempotent of the semigroup there is a maximal subgroup that is the group of symmetries of a subpolygon of $P$. In this paper we construct all of the maximal subgroups that can occur for any regular polygon $P$, and determine for which $P$ there exist nontrivial cyclic maximal subgroups, and for which there are only dihedral maximal subgroups.