2024/06/12 by Nicholson, John
#19B28 #20C05 #20C10 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2406.08692
We obtain a partial classification of the finite groups G for which the integral group ring ℤ G has projective cancellation, i.e. for which P ⊕ ℤ G ≅ Q ⊕ ℤ G implies P ≅ Q for projective ℤ G-modules P and Q. In particular, we determine when projective cancellation holds for a finite group with no exceptional binary polyhedral quotients. To do this, we prove a cancellation theorem based on a relative version of the Eichler condition. We then use a group theoretic argument to precisely determine the class of groups not covered by this result. The final classification is then obtained by applying results of Swan, Chen and Bley-Hofmann-Johnston which show failure of projective cancellation for certain groups.