Multiplicative induction and units for the ring of monomial representations

Let G be a finite group, and let A be a finite abelian G-group. For each subgroup H of G, Ω(H,A) denotes the ring of monomial representations of H with coefficients in A, which is a generalization of the Burnside ring Ω(H) of H. We research the multiplicative induction map Ω(H,A)→Ω(G,A) derived from the tensor induction map Ω(H)→Ω(G), and also research the unit group of Ω(G,A). The results are explained in terms of the first cohomology groups H1(K,A) for K≤G. We see that tensor induction for 1-cocycles plays a crucial role in a description of multiplicative induction. The unit group of Ω(G,A) is identified as a finitely generated abelian group. We especially study the group of torsion units of Ω(G,A), and study the unit group of Ω(G) as well.