Cayley graph of a group

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Definition

Let G be a group and S be a generating set for G. The Cayley graph of G with respect to S is defined as follows:

  • The vertex set of the graph is G.
  • Given two distinct vertices g,h \in G, there is an edge joining g to h if and only if g^{-1}h is in S \cup S^{-1}.

We typically consider the Cayley graph for a finitely generated group and a finite generating set of the group. Further, we can assume without loss of generality that S is a symmetric subset of G -- the inverse of any element of S is also in S.