Frobenius groups and classical maximal orders (Q2718316)

This monograph takes up, widens, and refines the ideas and material presented by the author and \textit{D. K. Harrison} [in J. Pure Appl. Algebra 126, No.~1-3, 51-86 (1998; Zbl 0912.20003)]. It starts out from a dictionary translating Frobenius groups with Abelian Frobenius kernel into certain maximal orders in finite dimensional central simple algebras \(A\) over Abelian extensions of \(\mathbb{Q}\). In fact, with each Frobenius complement \(G\) there is associated its so-called truncated group ring. Theorem: The truncated group ring of a Frobenius complement of order \(g\) is a maximal \(\mathbb{Z}[1/g]\)-order in some \(A\).NEWLINENEWLINENEWLINEThe monograph provides a complete computation of the truncated group rings of arbitrary Frobenius complements in terms of numerical invariants which determine the isomorphism classes of the Frobenius complements. Theorem: There is a natural bijection from the set of isomorphism classes of Frobenius groups with Abelian Frobenius kernel and with given Frobenius complement \(G\) to the set of \(\Aut(G)\)-orbits of the free Abelian semigroup on the set of all powers of maximal ideals of the center of the truncated group ring of \(G\). As a consequence, there are 569.342 isomorphism classes of such groups of order at most one million.NEWLINENEWLINENEWLINEA sharpening of Zassenhaus' theorem on the structure of solvable Frobenius complements is the classification theorem: Every Frobenius complement \(G\) has a unique normal subgroup \(N\) such that all Sylow subgroups of \(N\) are cyclic and \(G/N\) is isomorphic to 1, \(V_4\), \(A_4\), \(S_4\), \(A_5\) or \(S_5\). In terms of \(G\) and \(N\) the core index \([G:N]\) and the signature of \(G\) are defined. Theorem: A Frobenius complement is determined up to isomorphism by its order, core index, and signature.NEWLINENEWLINENEWLINEThe list of chapter headings may now help to realize the topics of this monograph. 2. Lemmas on truncated group rings. 3. Groups of real quaternions. 4. Proof of the classification theorem. 5.-10. Frobenius complements with core index 1, 4, 12, 24, 60, 120. 11. Counting Frobenius complements. 12. Maximal orders. 13. Isomorphism classes of Frobenius groups with Abelian Frobenius kernel. 14. Concrete constructions of Frobenius groups. 15. Counting Frobenius groups with Abelian Frobenius kernel. 16. Isomorphism invariants for Frobenius complements. 17. Schur indices and finite subgroups of division rings. This last chapter reflects the close relation of the presented method of constructing Frobenius groups to Amitsur's method of listing the finite subgroups of division rings.