Seminario Teoría de Grupos
The cancellation property for projective modules over integral group rings
Ponente: John Nichols (University of Glasgow)Fecha: miércoles 23 de octubre de 2024 - 11:30Lugar: Aula Naranja, ICMAT
Resumen:
Let G be a finite group and let Z[G] denote the integral group ring. If two finitely generated projective Z[G]-modules P and Q are isomorphic after taking a direct sum with the free module Z[G], are they necessarily isomorphic? If so, we say that Z[G] has the cancellation property. This was studied extensively in the 1960s-80s by H. Jacobinski, A. Fröhlich and R. G. Swan, and has applications both in number theory and algebraic topology. However, a complete classification of the finite groups which have the cancellation property had remained out of reach.
In this talk, I will present a new cancellation theorem for projective Z[G]-modules and explain how this leads to an approach to complete the classification using only finite computation. By utilising recent computer calculations of W. Bley, T. Hofmann and H. Johnston, this leads a classification of when cancellation occurs among all finite groups G which have no quotient which is a binary tetrahedral group, binary octahedral group, or a binary icosahedral group.
