On the existence of complements of residuals of finite group (Q1999796)

One of the interesting applications of the theory of formations is the search of sufficient conditions for a group to split over its \(\mathfrak{F}\)-residual subgroup. The most significant result in this direction is a famous theorem of Shemetkov: let \(\mathfrak{F}\) be a local formation; if the Sylow \(p\)-subgroups of \(G^{\mathfrak{F}}\) are abelian for every prime \(p\) dividing \(|G:G^{\mathfrak{F}}|\), then \(G^{\mathfrak{F}}\) has a complement in \(G\). The paper develops a possible generalization of this result. Let \(\omega\) be a set of primes and \(\mathfrak{F}\) an \(\omega\)-local Fitting formation. If a finite group \(G\) is generated by subnormal subgroups \(A_1,\dots,A_n\), the subgroups \(A^{\mathfrak{F}}_1,\dots,A^{\mathfrak{F}}_n\) are \(\omega\)-soluble and the Sylow \(p\)-subgroups of \(A^{\mathfrak{F}}_1,\dots,A^{\mathfrak{F}}_n\) are abelian for every \(p\in \omega\), then each \(\omega\mathfrak{F}\)-normalizer of \(G\) is an \(\omega\)-complement of \(G^{\mathfrak{F}}\) in \(G\).