Saturated formations and products of connected subgroups.
If \(\mathcal C\) is a non-empty class of groups, two subgroups \(A\) and \(B\) of a group \(G\) are said to be \(\mathcal C\)-connected if \(\langle a,b\rangle\in\mathcal C\) for all \(a\in A\) and \(b\in B\). If \(\pi\) and \(\rho\) are sets of primes, then \(S_\pi\) (\(S_\rho\)) denotes the class of finite soluble \(\pi\)-groups (\(\rho\)-groups) and \(S_\pi S_\rho\) the class of extensions of groups in \(S_\pi\) by groups in \(S_\rho\). If \(A\) and \(B\) are soluble subgroups of a finite group \(G\) such that \(G=AB\), then \(A\) and \(B\) are \(S_\pi S_\rho\)-connected if and only if \(O^\rho(B)\) centralises \(AO_\pi(G)/O_\pi(G)\), \(O^\rho(A)\) centralises \(BO_\pi(G)/O_\pi(G)\) and \(G\in S_{\pi\cup\rho}\) (Theorem 1). -- This result is then extended to so-called nilpotent-like Fitting formations of soluble groups (Theorem 2). These are non-empty saturated formations \(\mathcal F\) locally defined by a formation function \(f\) given by \(f(p)=S_{\pi(p)}\) for a set of primes \(\pi(p)\) containing \(p\), for each prime \(p\) belonging to the characteristic of \(\mathcal F\). Similar results hold for finite groups that are products of arbitrarily many pairwise permutable \(\mathcal F\)-connected \(\mathcal F\)-subgroups.
scientific article; zbMATH DE number 5987858
scientific article