Doctor of Philosophy
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. For non-p-groups, we introduce two other closely related invariants, the simple ghost number, which considers maps which are stably trivial when composed with any map from a simple module, and the strong ghost number, which considers maps which are ghosts after restriction to every subgroup of G. We produce the first computations of the ghost number for non-p-groups. We prove that there are close relationships between the three invariants, and make computations of the new invariants for many families of groups. We also discuss how computational algebra can be applied to calculate the ghost number.
Wang, Gaohong, "Ghost number of group algebras" (2014). Electronic Thesis and Dissertation Repository. 2231.