Abelian Frobenius kernels and modules over number rings
The paper provides an interesting example for playing off groups against modules (as, e.g., Abelian groups versus \(\mathbb{Z}\)-modules). Consider finite \(C_m\)-modules \(A\), where \(C_m\) is the cyclic group \(\langle e^{2\pi i/m}\rangle\). The action of \(C_m\) on \(A\) can be extended to a module action of \(R_m=\mathbb{Z}[e^{2\pi i/m},\tfrac 1m]\) on \(A\) iff \(C_m\) acts without fixed points. The non-zero finitely generated torsion modules \(A\) over \(R_m\), \(m\in\mathbb{N}\), are objects of a natural category which, by \(A\rightsquigarrow A\rtimes C_m\), is equivalent to the category \(\mathfrak F\) of finite groups which are either Abelian or metabelian Frobenius groups. The proof requires a careful study of the morphisms in the respective categories. \(\mathfrak F\) is a subcategory of the full subcategory \(\widetilde{\mathfrak F}\) of the category of groups whose objects are (i) groups \(G\) admitting an Abelian kernel \(N\) (i.e., \(N\vartriangleleft G\), \(G/N\) is locally finite, \(C_G(n)=N\) for \(1\neq n\in N\), \(N^m=N\) for all orders \(m\) of torsion elements in \(G/N\)), and (ii) subgroups of such \(G\), which intersect \(N\) trivially. Again, \(\widetilde{\mathfrak F}\) has a module theoretic analogue. For a group \(G\) denote by \(\mathbb{Z}_G\) the localization of \(G\) with respect to the orders of the torsion elements in \(G\), and by \(R_G\) the quotient \(\mathbb{Z}_GG/{\mathfrak a}_G\) with the ideal \({\mathfrak a}_G\) generated by the sums \(\sum_{h\in H}h\) for the finite subgroups \(H\) of \(G\). A truncated module is a pair \((M,G)\) with \(G\) locally finite, \(R_G\neq 0\) and a left \(R_G\)-module \(M\). They are the objects of a natural category which is equivalent to \(\widetilde{\mathfrak F}\). The proof rests on the three facts, (a) \(G\) is a generalized Frobenius complement iff \(R_G\neq 0\), (b) \(R_G\neq 0\Rightarrow G\) is finite or countable, (c) if \(N\) is an Abelian kernel of \(G\), then there exists a complement for \(N\) in \(G\). Within this set-up Frobenius groups can be translated into group ring modules, and the rings of coefficients are associated with algebraic number fields. This allows to apply number theory to group theory and is exploited for the calculation of the number of isomorphism classes of metabelian Frobenius groups with kernel of order \(n\) and complement of order \(m\).
scientific article; zbMATH DE number 1121691
scientific article