The cancellation property for projective modules over integral group rings

John Nicholson (University of Glasgow)

Wednesday 10th January 16:00-17:00 Maths 311B


Let G be a finite group and let Z[G] denote the integral group ring. If two 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. I will also report on attempts at completing these computations by myself and by W. Bley, T. Hofmann and H. Johnston.

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