Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group

Abstract

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors W G q ○ W H q ≅ W G × H q , where W G q denotes the q-deformed Witt–Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. To do this, we first establish an isomorphism of functors B G q ○ B H q ≅ B G × H q , where B G q denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H.