#### Vol. 1, No. 1, 2008

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

### Thomas L. Shelly and Janet E. Mills

Vol. 1 (2008), No. 1, 33–45
##### 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.

##### Keywords
semigroup, polygon, symmetries
Primary: 20M18
##### Milestones
Received: 11 June 2007
Accepted: 1 November 2007
Published: 28 February 2008

Communicated by Scott Chapman
##### Authors
 Thomas L. Shelly Department of Mathematics 901 12th Ave. P.O. Box 222000 Seattle, WA 98122-1090 United States Janet E. Mills Department of Mathematics 901 12th Ave. P.O. Box 222000 Seattle, WA 98122-1090 United States