Schreier coset graph

Definition

Suppose $G$ is a group, $H$ is a subgroup of $G$, and $A$ is a generating set for $G$. The Schreier coset graph of $H$ in $G$ is defined as follows:

1. Its vertices are the left cosets of $H$ in $G$.
2. Two vertices are adjacent if and only if there is an element of $A$ such that left multiplication by that element takes one of the cosets to the other. Note that we construct this as an undirected graph, even though the information also specifies a direction.

Note that some versions use right cosets. Switching from left to right cosets does affect the labels, but does not affect the isomorphism type of the graph, once we use that left and right coset spaces are naturally isomorphic (and a bit more checking, notably the fact that the undirected nature of the graph means that the Schrier coset graph for $A$ is the same as that for $A^{-1}$).

Examples

• If $H$ is the trivial subgroup of $G$, this just becomes the Cayley graph of $G$ for the generating set $A$.
• If $H$ is a normal subgroup of $G$, this can be identified withthe Cayley graph of the quotient group $G/H$ corresponding to the image of $A$ in $G/H$.