Permutability of subgroups and \(\mathfrak F\)-subnormality
General properties of formations inducing the WK-operator are studied. A subgroup \(H\) of a group \(G\) is called \({\mathfrak F}\)-composition subgroup (\({\mathfrak F}\) is a non-empty formation of finite groups), if there is a chain of subgroups \(G= H_0\geq H_1\geq\cdots\geq H_m=H\) such that, for every \(i\in\{1,2,\dots,m\}\), either \(H_i\) is normal in \(H_{i-1}\) or \(H^{\mathfrak F}_{i-1}\subseteq H\). The Wielandt-Kegel operator, or the WK-operator is the mapping \(r_{\mathfrak F}\colon G\to G^{\mathfrak F}\) that associates with each finite group \(G\) its \({\mathfrak F}\)-coradical if \(\langle H,K\rangle^{\mathfrak F}=\langle H^{\mathfrak F},K^{\mathfrak F}\rangle\) for arbitrary two \({\mathfrak F}\)-composition subgroups of \(G\). In this event, we also say that the formation \({\mathfrak F}\) induces the WK-operator. The main results are following theorems. Theorem 1. Let \({\mathfrak F}\) be a formation representable as \({\mathfrak F}={\mathfrak M}\times{\mathfrak H}\), where \(\pi({\mathfrak M})\cap\pi({\mathfrak H})=\emptyset\), \({\mathfrak M}^2={\mathfrak M}\) is a nonempty \(S\)-closed formation, \({\mathfrak H}=\bigtimes_{i\in I}{\mathfrak S}_{\pi_i}\), and moreover \(\pi_\ell\cap\pi_k=\emptyset\) for all \(k\neq\ell\) in \(I\). Then the formation \({\mathfrak F}\) induces the WK-operator. Theorem 2. Let \(f\) be a canonical formation function of \({\mathfrak F}\) inducing the WK-operator. Let \(H\) and \(K\) be \({\mathfrak F}\)-composition subgroups of \({\mathfrak I}=\langle H,K\rangle\); moreover, all Abelian composition factors of \(H\) belong to \({\mathfrak F}\). Then \(HK=KH\) if and only if, for all \(p\) in \(\pi({\mathfrak F})\), each homomorphism from \({\mathfrak I}\) into an \(f(p)\)-group takes \(H\) and \(K\) to permutable groups.
scientific article; zbMATH DE number 1023908
scientific article