Brauer relations for finite groups in the ring of semisimplified modular representations

Matthew Spencer

Published 2016-10-24Version 1

We study the kernel of the map between the Burnside ring of a finite group $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$ modules for a prime $p$, which takes a $G$-set to its associated permutation module. We are able to give a classification of the primitive quotient of the kernel; the kernel for $G$ modulo elements coming from the kernel for proper subquotients of $G$, for all finite groups. In the soluble case we are able to identify exactly which groups have primitive elements in the kernel and we give generators for the primitive quotient.