On complements of coradicals of finite groups (Q2821897)

Let \(G\) be a finite group, \(\mathfrak F\) a formation of groups. The classical theorem of \textit{L. A. Shemetkov} [Mat. Sb., N. Ser. 94(136), 628--648 (1974; Zbl 0323.20036)] states that if \(\mathfrak F\) is local and all \(p\)-Sylow subgroups of \(G\) are abelian in case \(p\) divides the index of the coradical \(G^{\mathfrak F}\), then \(G^{\mathfrak F}\) has a complement. The authors, applying deliberate arguments, strengthen this result for \(\omega\)-local formations investigated by the authors in [Discrete Math. Appl. 11, No. 5, 507--527 (2001; Zbl 1057.20013); Math. Notes 71, No. 1, 39--55 (2002; Zbl 1068.20024)]. An \(\omega\)-local formation is \(\omega \mathrm{LF}(f)=\{G \mid G/O_{\omega}(G)\in f(\omega'),\,G/F_{p}(G)\in f(p)\,(p\in\omega\cap\pi(G))\}\), where \(\omega\subseteq\mathbb{P}\), \(\omega'=\overline{\omega}\), \(f\:\omega\cup\{\omega'\}\to\{\text{formations}\}\), \(\delta(p)={\mathfrak S}_{p'}{\mathfrak N}_p\).NEWLINENEWLINESet \({\mathfrak F}=\omega \mathrm{LF} (f)\). The first main result of the paper, yielding many corollaries, is as follows. If \(\omega_1\neq\emptyset\) is the set of all \(p\in\omega\) for which \(G^{\mathfrak F}\) has an abelian \(p\)-Sylow subgroup, then \(G^{\mathfrak F}\) has an \(\omega_1\)-complement in any extension of \(G\) (that is, the intersection is an \(\omega_1'\)-group). The second main result of the paper, also yielding numerous corollaries, studies \(\omega{\mathfrak F}\)-normalizers \(H\), that is subgroups of \(G\) with \(H/\Phi(H)\cap O_{\omega'}(H)\in{\mathfrak F}\) and there is a maximal chain of \(G\) starting from \(G\), ending in \(H\), with each term \(\mathfrak F\)-critical in the preceding one. If \(\mathfrak F\) is Fitting, \(G\) is a product of subnormal subgroups with \(\mathfrak F\) coradical \(\omega\)-soluble and possessing an abelian Sylow \(p\)-subgroup for all \(p\in\omega\), then any \(\omega{\mathfrak F}\)-normalizer is an \(\omega\)-complement of \(G^{\mathfrak F}\) in \(G\).