Abstract
In this chapter we examine some interactions between graphs and groups. Some problems concerning groups are best attacked by using graphs. Although in this context a graph is hardly more than a visual or computational aid, its use does make the presentation clearer and the problems more manageable. The methods are useful both in theory and in practice: they help us to prove general results about groups and particular results about individual groups. The first section, about Cayley and Schreier diagrams, illustrates well both these aspects. It also contains an informal account of group presentations.
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Notes
There is a vast literature concerned with group presentations, including the use of Cayley and Schreier diagrams. The basic book is perhaps W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Second Edition, Dover, New York, 1976;
the connections with geometry are emphasised in H. S. M. Coxeter, Regular Complex Polytopes, Cambridge University Press, New York, 1974.
Numerous articles deal with the computational aspect, in particular J. A. Todd and H. S. M. Coxeter, A practical method for enumerating cosets of a finite abstract group, Proc. Edinburgh Math. Soc. (2) 5 (1936) 26–34,
which was the first paper in this line and J. Leech, Computer proof of relations in groups, in Topics in Group Theory and Computation (M. P. J. Curran, ed.), Academic Press, New York, 1977, in which some more recent developments are described.
Max Dehn posed the word problem in Über unendliche diskontinuierliche Gruppen, Math. Ann. 71 (1911) 116–144, and gave the above discussed presentation of the group of the trefoil in Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910) 137–168.
The word problem was shown to be intrisically connected to logic by P. S. Novikov, On the algorithmic unsolvability of the word problem, Amer. Math. Soc. Transl. (2) 9 (1958) 1–22
and G. Higman, Subgroups of finitely presented groups, Proc. Royal Soc. A, 262 (1961) 455–475.
For the properties of knots and their groups the reader is referred to R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 57, Springer-Verlag, New York, 1977.
An exposition of matrix methods in graph theory can be found in N. Biggs, Algebraic Graph Theory, Cambridge University Press, New York, 1974.
The first striking result obtained in this way, Theorem 9, is due to A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Dev. 4 (1960) 497–504.
The connection between graphs and sporadic simple groups is discussed in detail in R. Brauer and C. H. Sah (editors), Theory of Finite Groups: a Symposium, Benjamin, Menlo Park, 1969. Since then however several more sporadic groups have been found.
The fundamental enumeration theorem of G, Pólya appeared in Kombinatorische Anzahlbestimmungen für Gruppen und chemische Verbindungen, Acta Math. 68 (1937) 145–254.
Many enumeration techniques were anticipated by J. H. Redfield, The theory of group-reduced distributions, Amer. J. Math. 49 (1927) 433–455.
The standard reference book for Pólya-type enumeration is F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, New York, 1973.
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Bollobás, B. (1979). Graphs and Groups. In: Graph Theory. Graduate Texts in Mathematics, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-9967-7_8
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