  The critical group of a graph is a finite abelian group whose order is the
number of spanning forests of the graph. This paper provides three basic
structural results on the critical group of a line graph.
  The first deals with connected graphs containing no cut-edge. Here the number
of independent cycles in the graph, which is known to bound the number of
generators for the critical group of the graph, is shown also to bound the
number of generators for the critical group of its line graph.
  The second gives, for each prime p, a constraint on the p-primary structure
of the critical group, based on the largest power of p dividing all sums of
degrees of two adjacent vertices.
  The third deals with connected graphs whose line graph is regular. Here known
results relating the number of spanning trees of the graph and of its line
graph are sharpened to exact sequences which relate their critical groups.
  The first two results interact extremely well with the third. For example,
they imply that in a regular nonbipartite graph, the critical group of the
graph and that of its line graph determine each other uniquely in a simple
fashion.
