Factorizable groups and formations. (Q2386437)

Let \(\mathcal F\) be a saturated formation of finite groups that is locally defined by the formation function \(f\) (called ``local satellite'' of \(\mathcal F\)), i.e., \({\mathcal F}=\text{LF}(f)\). Such a satellite is called semi-integrated if, for each prime \(p\), either \(f(p)\subseteq\mathcal F\) or \(f(p)=\mathcal E\), the class of all groups. The central concept of the paper is the following: an element \(x\) of a group \(G\) is called a \(Z_{f,G}\)-element if there exists an \(f\)-central chief factor \(H/L\) of \(G\) such that \(x\in H\setminus L\). Using this definition, the authors prove the following main results: Suppose that \({\mathcal F}=\text{LF}(f)\), where \(f\) is semi-integrated and let \(G\) be a group. Denote by \(\omega\) the set of all primes \(p\) such that \(f(p)\neq\mathcal E\). (i) If, for each \(p\in\omega\), every \(x\in G_p\setminus\Phi(G_p)\) is a \(Z_{f,G}\)-element, where \(G_p\) is a Sylow \(p\)-subgroup of \(G\), then \(G\in\mathcal F\) (Theorem 3.1). (ii) If, for each \(p\in\omega\), \(G\) possesses Abelian Sylow \(p\)-subgroups and every cyclic \(p\)-subgroup complemented in a Sylow \(p\)-subgroup of \(G\) is generated by a \(Z_{f,G}\)-element, then \(G\in\mathcal F\) (Theorem 3.2).