  Motivated by Freyd's famous unsolved problem in stable homotopy theory, the
generating hypothesis for the stable module category of a finite group is the
statement that if a map in the thick subcategory generated by the trivial
representation induces the zero map in Tate cohomology, then it is stably
trivial. It is known that the generating hypothesis fails for most groups.
Generalizing work done for $p$-groups, we define the ghost number of a group
algebra, which is a natural number that measures the degree to which the
generating hypothesis fails. We describe a close relationship between ghost
numbers and Auslander-Reiten triangles, with many results stated for a general
projective class in a general triangulated category. We then compute ghost
numbers and bounds on ghost numbers for many families of $p$-groups, including
abelian $p$-groups, the quaternion group and dihedral $2$-groups, and also give
a general lower bound in terms of the radical length, the first general lower
bound that we are aware of. We conclude with a classification of group algebras
of $p$-groups with small ghost number and examples of gaps in the possible
ghost numbers of such group algebras.
