Radicals commuting with cartesian products

Abstract.

For any radical R of abelian groups which does not commute with arbitrary cartesian products we define the norm $\| R\|$ to be the least cardinal for which there exists a family, of this size, of groups $G_\alpha$ such that $R\prod G_\alpha \ne \prod R G_\alpha$. This norm $\| R\|$ is always regular. Assuming GCH, we construct reduced products G to show that every regular cardinal $\kappa$ which is not greater that any weakly compact cardinal is the norm of a suitable group radical R_G.