nLab
Cayley graph

Contents

Contents

Idea

The Cayley graph (also called the Cayley quiver) of a group, GG with a given set of generators, XX, encodes how the chosen generating elements operate (by multiplication) on the elements of the group.

Definition

Given a group, GG, and a set of generators, XX, which we regard as a subset of the underlying set |G||G| of GG, we can form a directed graph with the elements of GG as the vertices and with its edges labelled (some people say ‘coloured’) by the elements of XX with an edge joining a vertex, gg, to the vertex gxg x labelled by xx

gxgx g \overset{x}{\longrightarrow} g x

This graph is called the Cayley graph of the group, GG, relative to the set of generators.

Example

Consider one of the standard presentations of S 3S_3, (a,b:a 3,b 2,(ab) 2) (a,b : a^3, b^2, (a b)^2). Write r=a 3 r = a^3, s=b 2s = b^2, t=(ab) 2t = (ab)^2.

The Cayley graph is easy to draw. There are two triangles corresponding to 1aa 21 \to a \to a^2 and to its translate by bb, baba 2bb \to a b \to a^2 b, flipping the orientation of the second, and three 2-cycles, 1b11\to b\to 1, aabaa\to a b\to a and a 2a 2ba 2a^2\to a^2b\to a^2.

To understand the geometric significance of this graph we compare the algebraic information in the presentation with the homotopical information in the graph (considered as a CW-complex).

Looking at the presentation it leads to a free group, FF, on the generators, aa and bb, so FF is free of rank 2, but the normal closure of the relations N(R)N(R) is a subgroup of FF, so it must be free as well, by the Nielsen-Schreier theorem. Its rank will be 7, given by the Schreier index formula.

Looked at geometrically, this will be the fundamental group of the Cayley graph, of the presentation. This group is free on generators corresponding to edges outside a maximal tree, and, of course, there are 7 of these.

References

General

See also:

Distance on Cayley graph

On distances in Cayley graphs:

For symmetric groups (permutations):

  • Mohammad Hossein Ghaffri, Zohreh Mostaghim, Distance in Cayley graphs on permutations generated by kmk m cycles, Transactions on Combinatorics, Vol 6 No. 3 (2017) (pdf)

Last revised on February 9, 2020 at 06:29:27. See the history of this page for a list of all contributions to it.