Radicals commuting with cartesian products
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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 RG.
KeywordsAbelian Group Group Radical Arbitrary Cartesian Product Compact Cardinal Suitable Group
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