  The critical group of a graph is a finite abelian group whose order is the
number of spanning forests of the graph. For a graph G with a certain
reflective symmetry, we generalize a result of Ciucu-Yan-Zhang factorizing the
spanning tree number of G by interpreting this as a result about the critical
group of G. Our result takes the form of an exact sequence, and explicit
connections to bicycle spaces are made.
