Abstract
This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See e.g. Parsons (2008).
Referring to graphs.
In group theory there is a terminological distinction (although not always a very sharp one) between ‘representation’ and ‘presentation’. Namely, a representation of a group describes an abstract group in terms of linear transformations of vector spaces (Fulton and Harris 1991). But there is also a more general use of the term as any ‘description’ of a group as a group of transformations of some mathematical object. Presentation is the way to define a group by generators and their relations (Skornjakov 1990; Johnson 1990). In this respect, given that the structural skeleton of a particular CG is determined by a chosen set of generators, it would be rather called ‘presentations’ of groups. But sometimes, to emphasise the graphic nature of CG, it is said, and this is the sense I use, that CGs are representations (in the sense of descriptions) of groups by means of graphs. The distinction between the two notions may be interesting from the epistemological perspective: a representation presupposes some background of other type of mathematical objects and kind of modelling by means of those objects, whereas presentation determines a group by means of ‘intrinsic’ properties.
Sometimes called group ‘multiplication’.
In algebra such concatenations are read from the right to the left.
Multiplication here means application of a group operation (in this example it is addition).
In fact, one can start with any element of the group, but it is easier to start with the identity element.
Except Lie groups and groups of topological spaces.
This term is used for example in Feferman (2000).
For Cayley theorem see Cayley (1854).
A vertex-transitive graph is a graph such that every pair of vertices is equivalent under some element of its automorphism group. More explicitly, a vertex-transitive (Eric Weisstein) graph is a graph whose automorphism group is transitive. This is the definition cited by MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Vertex-TransitiveGraph.html from (Holton and Sheehan 1993, p. 27).
There are other geometries of groups, for example, Lie groups considered as manifolds, but we are here concerned only about infinite groups with finite sets of generators, which are studied by means of CGs.
Word length problem allowing to change initial alphabet by using new letters conjugate to generators recently appeared in some physics-based theories, e.g. Majumdar et al. (2009).
If A and B are two finite sets of generators of the group G then the identity mapping between the metric spaces \( G,d_{W}^{A} \) and \( G,d_{W}^{B} \) is quasi-isometry, i.e. the word metric is unique up to quasi-isometry.
Examples of such properties of finitely generated groups include: the growth rate of a finitely generated group (the growth rate counts the number of elements that can be written as a product of length n, e.g. polynomial and exponential growth; a free group with a finite rank k > 1 has an exponential growth rate), the number of ends of a group, hyperbolicity of a group, amenability (roughly measurability) of a finitely generated group, being virtually abelian (that is, having an abelian subgroup of finite index), being virtually free, being finitely presentable, being a finitely presentable group with solvable word problem (problem of deciding whether two words represent the same element), and others.
It is important to note that the length of a word depends on the generating set. For example, in the group \({\mathbb{Z}}\otimes{\mathbb{Z}}\) for a set of generators S = {(1, 0), (0, 1), (−1, 0), (0, −1)} the length of (1, 1) equals 2, but for S = {(1, 1), (0, 1), (−1, −1), (0, −1)} the length of the same (1, 1) equals 1.
Gromov (1993, pp. 8–10).
This notation is from Gromov (1993, p. 8).
Ibid, p. 9.
Ibid, p. 10.
References
Cayley A (1854) On the theory of groups as depending on the symbolic equation θn = 1. Philos Mag 7(4):40–47
Feferman S (2000) Mathematical Intuition vs. Mathematical Monsters. Synthese 125(3):317–332
Fulton W, Harris J (1991) Representation theory. A first course, graduate texts in mathematics, readings in mathematics, vol 129. Springer, New York, pp 1–3
Gromov M (1993) Asymptotic invariants of infinite groups. In ‘Geometric Group Theory’, Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, pp 1–295
Grossman I, Magnus W (1964) Groups and their graphs. MAA, pp 44–56
Gvozdjak P, Sirán J (1994) Arc-transitive non-Cayley graphs from regular maps. Acta Math Univ Comenianae LXIII(2):309–313
Heinzmann G (2007) Arithmetic and cognitive evidence: a pragmatic approach’, section c.1 philosophy of mathematics and logic. LMPS, Beijing
Holton DA, Sheehan J (1993) The Petersen graph. Cambridge University Press, Cambridge
Johnson DL (1990) Presentations of groups. Cambridge University Press, Cambridge
Krön B (2008) ‘Cutting up graphs revisited’, forthcoming
Majumdar A, Robbins JM, Zyskin M (2009) `Tangent unit-vector fields: nonabelian homotopy invariants and the dirichlet energy’, Compte Rendus Mathématiques (accepted)
McKay BD, Royle G (1990) Construction of all simple graphs with at most 26 vertices and transitive automorphism group. Ars Combin 30:161–176
Parsons Ch (2008) Mathematical thought and its objects. Cambridge University Press, Cambridge, pp 138–185
Seress Á (2005) Square-free non-Cayley numbers. On vertex-transitive non-Cayley graphs of square-free order. Designs, Codes and Cryptography, 34, Numbers 2–3, February, 265–281
Skornjakov L (1990) General algebra, vol 2. Nauka, Moscow ch.1.3
Watkins ME (1990) Vertex-transitive graphs that are not Cayley graphs. In: Hahn G et al (eds) Cycles and rays. Kluwer, The Netherlands, pp 243–256
Wise D (2008) http://www.math.mcgill.ca/wise/ggt/cayley.html. McGill University, Canada
Acknowledgments
I am grateful to Andrey Rodin, Bernhard Krön, Leon Horsten, John Mayberry, Maxim Zyskin, Michael Rudnev, Robin Brown, Tim Riley, and my supervisor Hannes Leitgeb for their useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Starikova, I. Why do Mathematicians Need Different Ways of Presenting Mathematical Objects? The Case of Cayley Graphs. Topoi 29, 41–51 (2010). https://doi.org/10.1007/s11245-009-9065-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11245-009-9065-4