A Group Portrait on a Surface of Genus Five

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. Introduction and Historical Perspective A group is a set and an associative binary operation which contains an identity such that each element has an inverse element in the group. Therefore, a group is an abstract object. Groups were originally thought of as permutations of some other mathematical structure, such as a set of points. This permutation group idea comes very naturally from the set of symmetries of physical objects. Thus the symmetries of an equilateral triangle are a group with six elements. It follows that groups can be both abstract objects and real physical motions of a symmetric object. A group can also be thought of as a set of transformations of a "plane" into itself that is closed under composition. Some groups of transformations can be realized on the Euclidean plane and these give rise to the tessellations of the plane. Many groups need to be realized on the hyperbolic plane and give rise to tessellations of it. A finite group is a finite set of transformations which is closed under composition. If a finite group were represented on an infinite plane, then the fundamental region would have to be infinite. Therefore, a finite group is represented as a set of transformations of a compact two dimensional surface, such as the sphere or the torus. These surfaces must become more complex in order to contain the portraits of some groups. This paper is part of a series of papers on how to draw a portrait of a finite group, in the style of Burnside [1]. Burnside started with circles in the plane and used inversion in the circle as the transformation. The relationship between circles determines the group Figure 1, Portrait of a Free Group generated by these transformations. When we look at the regions this generates inside of a circle we get Figure 1. Burnside [1, p. 379] constructed a free group on 2 generators, F2, using two mutually tangent circles and the line tangent to them (a circle centered at infinity). We only need to picture the part of each region inside the circle through the three points of tangency. Like Burnside, the initial region is the "triangle" labeled E and its corresponding shaded region. Each “triangle” is bounded by arcs colored red, blue or black in our sketch. Inversion in any single arc will take a shaded region into a non-shaded region and viceversa. Each region can be labeled by the transformation needed to get from E to that region. Since we are interested in orientation-preserving transformations, each group action is represented by the composite of two such inversions. Inversion through first a red arc and then a blue arc corresponds to multiplying on the left by the generator S. Multiplying on the left by the generator T corresponds to inversion through black and then red. Multiplying on the left by ST corresponds to inversion through black and then blue. If we considered inversion through a black arc first and then a blue arc as the inverse of a single generator, R, then we could interpret this picture as a portrait of a group with presentation 1 | , , 2 RST T S R T . This construction fills up a unit disk with black and white regions and the transformations are given in the same way. We have used Geometer’s SketchPad [4] to reconstruct this portrait of a free group on two Figure 2 The Fundamental Region for SG(32,2) generators (Figure 1), similar to a figure in Burnside [1, p. 380]. Now suppose that we have a finite group, G, generated by 2 generators. The group G is the image of F2 by a normal subgroup N. Specifically, two strings of generators represent the same element of the group if the product of one string and the inverse of the other string is in the subgroup N. One example of this is that a rotation of 120o clockwise is the same as two rotations of 120o counterclockwise. After associating an element of F2 to each region, the final step is to identify all regions with labels from the subgroup N. After this identification, we have the finite group G. However the circle in Figure 1 still has an infinite number of regions labeled with elements from the presentation 2 T . This circle can be thought of as the Poincare disk model of hyperbolic space. At this point, we choose a connected set of regions which contains a single region for each group element and whose label corresponds to that group element. This set of regions is called the Fundamental Region for the finite group. Any region outside of the fundamental region is equivalent to some region within the fundamental region. Therefore, the compact two dimensional surface is derived by folding the fundamental region in certain ways. This is the same idea as constructing a torus by taking a rectangle and identifying the top and bottom as connecting to each other and the two sides as connecting. In the diagram in Figure 1, each element has infinite order and the curvilinear triangles get smaller and smaller as they approach the boundary of the circle. This means that the connected region in Figure 1 would have a ragged boundary with parts of the boundary intersecting each point. This can be fixed by picturing a different tiling of the hyperbolic plane or by drawing a polygonal region with the same relationship between the triangles. The second approach has the advantage that the triangles remain easily visible as they get near the boundary of the polygonal region and this portrait is pictured in Figure 2. The portraits developed are topologically equivalent to the model that we want, but even the areas of the regions are changed. Compact surfaces are classified topologically by genus and orientability. Every compact orientable surface with genus g is topologically equivalent to a sphere with g handles. Very roughly, the genus is the number of "donut" holes that a surface has. This is why a donut and a coffee cup are topologically equivalent. Thus, every compact surface of genus g may be constructed in many different, but topologically equivalent ways. We will pick a symmetric way of representing the surface and use it. This surface Figure 3 Connectivity of the Fundamental Region may be drawn and colored with each face composed of one white and one black region. These faces represent a finite group of transformations, which act on the surface in the style of Burnside [1]. The choice of surface is made arbitrarily with the correct orientability and genus. There have been several Bridges papers which pictured regular maps on surfaces of genus 3 to 7 (for example [3] and [7]) and their associated tilings and the related topic of portraits of groups using techniques in this paper ([9] and [10]). The automorphism groups of these tilings are the groups PSL(2,7) (in [3]) and S5 (in [7]). The portraits in [9] are of the dicyclic group of order 12 and the group of order 16 with notation < 2, 2 | 2> (see [2, p. 134]) acting as orientation-preserving transformations and the group P48 = SG(48, 33) pictured as a group where some transformations are orientation-reversing.