On characters and fixed points of coprime operator groups
Suppose that \(A\) is a finite group of operators on a finite soluble group \(G\) and that \((|G|,|A|)=1\). The main theorem (Theorem A) states that if \(|C_G(A)|\leq n\), then \(|G:F_{2\ell+1}(G)|\) is bounded by a function of \(n\) and \(|A|\), where \(\ell\) is the length of the longest chain of subgroups of \(A\). Two corollaries for locally finite groups are given. The proof of this uses a theorem of Turull and the following result on representations (Theorem B). For any finite group \(A\), there exists \(\varepsilon>0\) such that the multiplicity in \(V_A\) of any of its irreducible components is at least \(\varepsilon(\dim V)\), where \(V\) is an irreducible \(kAG\)-module, \(k\) is a field the characteristic of which does not divide \(|A|\) and \(G\) is a finite, soluble group on which \(A\) acts and \((|G|,|A|)=1\). Now suppose that \(A\) is any finite group and that the characteristic of \(k\) does not divide \(|A|\). Given \(\varepsilon>0\), let \(\Delta(\varepsilon)\) denote the class of \(kA\)-modules \(W\) having the property that the multiplicity in \(W\) of any of its irreducible components is at least \(\varepsilon(\dim W)\). To prove Theorem B, a number of results about \(\Delta(\varepsilon)\) are obtained, among them Theorem C: Given \(\varepsilon>0\), there exists \(\delta>0\) such that any tensor product of modules in \(\Delta(\varepsilon)\) lies in \(\Delta(\delta)\). In this notation, Theorem B states that \(V_A\in\Delta(\varepsilon)\); this is proved by reducing it to the case when \(k=\mathbb{C}\) and then working with characters. The crucial step is to factorize the character of \(V_A\) into a number of characters, each of modules in \(\Delta(\varepsilon_0)\), where \(\varepsilon_0=\varepsilon_0(A)\), and then to use Theorem C. The factorization theorem has the corollary that if \(\chi\) is a character of the semidirect product \(AG\) and \(\chi_A\) is irreducible, then there exists an \(A\)-group \(E\) and an irreducible character \(\psi\) of \(AE\) such that \(\psi_E\) is irreducible, \(\psi_A=\chi_A\) and \(E\) is a direct product of extraspecial groups. This factorization theorem is proved first, then the theorems about \(\Delta(\varepsilon)\), leading to a generalization of Theorem C. The final section is devoted to the proof of Theorem B.
scientific article; zbMATH DE number 4152686