Schreier coset graph

In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set S={s_i : i in I} of G, and a subgroup H ≤ G. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset.

The graph is named after Otto Schreier, who used the term "Nebengruppenbild".^[1] An equivalent definition was made in an early paper of Todd and Coxeter.^[2]

Description

The Schreier graph of a group G, a subgroup H, and a generating set S⊆G is denoted by Sch(G,H,S) or Sch(H\G,S). Its vertices are the right cosets Hg = {hg : h in H} for g in G, and its edges are of the form (Hg, Hgs) for g in G and s in S.

More generally, if X is a G-set, the Schreier graph of the action of G on X (with respect to S⊆G) is denoted by Sch(G,X,S) or Sch(X,S). Its vertices are the elements of X, and its edges are of the form (x,xs) for x in X and s in S. This includes the original Schreier coset graph definition, as H\G is a naturally a G-set with respect to multiplication from the right. From an algebraic-topological perspective, the graph Sch(X,S) has no distinguished vertex, whereas Sch(G,H,S) has the distinguished vertex H, and is thus a pointed graph.

The Cayley graph of the group G itself is the Schreier coset graph for H = {1_G} (Gross & Tucker 1987, p. 73).

A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma (Conder 2003).

The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids. A subgroup H of a group G determines a covering morphism of groupoids ${\displaystyle p:K\rightarrow G}$ and if S is a generating set for G then its inverse image under p is the Schreier graph of (G, S).

Applications

The graph is useful to understand coset enumeration and the Todd–Coxeter algorithm.

Coset graphs can be used to form large permutation representations of groups and were used by Graham Higman to show that the alternating groups of large enough degree are Hurwitz groups, (Conder 2003).

Stallings' core graphs^[3] are retracts of Schreier graphs of free groups, and are an essential tool for computing with subgroups of a free group.

Every vertex-transitive graph is a coset graph.

References

1. Schreier, Otto (December 1927). "Die Untergruppen der freien Gruppen". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 5 (1): 161–183. doi:10.1007/BF02952517.
2. Todd, J.A; Coxeter, H.S.M. (October 1936). "A practical method for enumerating cosets of a finite abstract group". Proceedings of the Edinburgh Mathematical Society. 5 (1): 26–34. doi:10.1017/S0013091500008221.
3. John R. Stallings. "Topology of finite graphs." Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565

• Magnus, W.; Karrass, A.; Solitar, D. (1976), Combinatorial Group Theory, Dover
• Conder, Marston (2003), "Group actions on graphs, maps and surfaces with maximum symmetry", Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser., vol. 304, Cambridge University Press, pp. 63–91, MR 2051519
• Gross, Jonathan L.; Tucker, Thomas W. (1987), Topological graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons, ISBN 978-0-471-04926-5, MR 0898434
• Schreier graphs of the Basilica group Authors: Daniele D'Angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda
• Philip J. Higgins, Categories and Groupoids, van Nostrand, New York, Lecture Notes, 1971, Republished as TAC Reprint, 2005