On the critical group of the Möbius ladder graph

The critical group of a connected graph is a finite abelian group whose order is the number of spanning trees and whose structure is a subtle isomorphism invariant of the graph. In this paper we study the structure of the critical group on the Möbius ladder, and we prove that the Smith normal form of the critical group is not cyclic but is always the direct sum of two or three cyclic groups.