A Group Portrait on a Surface of Genus Five

Jay Zimmerman
Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture (2009)
Pages 259–264


This paper represents a finite group with 32 elements as a group of transformations of a compact surface of genus 5. In particular, we start with a designated pair of regions of this surface, and each region is labeled with the group element, which transforms the designated region into it. This gives a portrait of that finite group. These surfaces and the regions corresponding to the group elements are shown in this paper. William Burnside first gave a simple example of such a portrait in his 1911 book, “Theory of Groups of Finite Order”. This paper is the third paper in a series which models groups as groups of transformations on a compact surface in the style of William Burnside.