Cayley graph

Definition

Let $G$ be a group and let $S$ be a generating set of $G$. The Cayley graph $\Gamma(G, S)$ is the directed graph with vertex set $G$. For each $g \in G$ and $s \in S$, it has an edge with colour $s$ from $g$ to $g s$.

Remarks

For $G = \mathbb{Z}$ and $S = \{1, -1\}$, the Cayley graph is an infinite path. For $\mathbb{Z}/n\mathbb{Z}$ with a generator and its inverse, it is the cycle $C_n$. The Cayley graph of a free group with respect to a basis is a tree. The left multiplication action of $G$ on itself acts on $\Gamma(G, S)$, so each Cayley graph is vertex-transitive. Arthur Cayley first considered these graphs for finite groups in 1878.

Explanations

L1 human

A Cayley graph draws a group as a network. Each element is a point, and each generator gives an arrow from $g$ to $g s$. Paths in the graph correspond to words in the generators.