Ghost numbers of group algebras II

J. Daniel Christensen, Gaohong Wang

Published 2013-10-21, updated 2015-02-20Version 2

We study several closely related invariants of the group algebra $kG$ of a finite group. The basic invariant is the ghost number, which measures the failure of the generating hypothesis and involves finding non-trivial composites of maps each of which induces the zero map in Tate cohomology ("ghosts"). The related invariants are the simple ghost number, which considers maps which are stably trivial when composed with any map from a suspension of 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, e.g., for the dihedral groups at all primes, as well as many new bounds. We prove that there are close relationships between the three invariants, and make computations of the new invariants for many families of groups.