2017/06/07 by Margolis, Leo
#16S34 #16U60 #20C05 #20C11 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1706.02117
A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let R be a commutative ring, G a finite group, N a normal p-subgroup of G and denote by RG the group ring of G over R. It is shown that a torsion unit u in ℤG mapping to the identity under the natural homomorphism ℤG → ℤG/N is conjugate in the unit group of ℤpG to an element in N. Here ℤp denotes the p-adic integers. The result is achieved proving a result in the context of the so-called double action formalism for group rings over p-adic rings. This widely generalizes a theorem of Hertweck and a related theorem by Caicedo-Margolis-del Río and has consequences for the study of the Zassenhaus Conjecture for integral group rings.