Irreducible disconnected systems in groups
Let \(G\) be an arbitrary group with subgroup \(A\). The index \([A:A\cap A^g]\) (\(g\in G\)) is called a subdegree of the pair \((A,G)\). This concept was presented by \textit{I. M. Isaacs} and \textit{C. E. Praeger} [J. Algebra 159, No. 1, 158-175 (1993; Zbl 0815.20002)]. Let \(D\) denote the set of all subdegrees of \((A,G)\). Assume further that \(A\) is not normal in \(G\) and all the subdegrees of \((A,G)\) are finite. Then the common divisor graph \(\Gamma\) is associated to \((A,G)\): its vertex set is \(D-\{1\}\), and two different subdegrees \(s,t\) are joined by an edge if \(\gcd(s,t)\not=1\). In the above mentioned paper, it is proved that \(\Gamma\) has at most two connected components. Assume that \(\Gamma\) is disconnected. Let \(D_1\) be the component of \(\Gamma\) containing \(\min(D-\{1\})\). In [J. Algebra 193, No. 2, 616-628 (1997; Zbl 0891.20018)], the author proved that if \(A\) is stable in \(G\) (a property which holds when \(A\) or \([G:A]\) is finite), then the set \(H=\{g\in G\mid[A:A\cap A^g]\in D_1\}\cup\{1\}\) is a subgroup of \(G\). In this case he says that \(A<H<G\) is a disconnected system (briefly: a system) of \(G\). In this paper, the author deals with some fundamental types of systems. A system \(A<H<G\) is irreducible if there does not exist a system of the form \(AN/N<H/N<G/N\), where \(1<N\triangleleft G\), \(AN<H\). Theorem A gives restrictions on the finite nilpotent normal subgroups of \(G\), when \(G\) possesses an irreducible system. In particular, if \(G\) is finite then the Fitting subgroup \(\text{Fit}(G)\) of \(G\) is primary. The author considers also general systems. Corollary 4.2. says that if \(G\) is finite and \(G\) possesses a system \(A<H<G\) then \(G\) has a proper normal subgroup \(N\) such that \(AN/N\) is not nilpotent and \(\text{Fit}(G/N)\) is primary. Theorem B says that for any system \(A<H<G\), \(N_G(N_G(A))=N_G(A)\). Theorem C and Corollary C' generalize Theorem 2 from a paper of \textit{C. E. Praeger} [Mathematics Research Report, Aust. Natl. Univ. (1989)].
scientific article; zbMATH DE number 1356477