Some notes on the minimal subgroups of Fitting subgroups of finite groups
Many authors have studied the structure of a finite group \(G\), under the assumption that all minimal subgroups of \(G\) are well-situated in the group. First results in this line are due to Itô and Buckley. Recently, \textit{Y. Wang} and the reviewer [J. Pure Appl. Algebra 153, No. 2, 121-127 (2000; Zbl 0967.20009)] studied the influence of c-normality of the minimal subgroups on the structure of \(G\) through the theory of formations and obtained many new results which generalize some classical and previous ones. The present paper represents an attempt to extend and improve the results of the above mentioned authors by studying the influence of the minimal subgroups of the Fitting subgroup of \(G\). They consider saturated formations \(\mathcal F\) containing the class of nilpotent groups and look for conditions that guarantee the group \(G\) is in \(\mathcal F\) if a fixed quotient \(G/H\), where \(H\) is a normal subgroup of \(G\) enjoying certain embedding properties concerning c-normality and \(\mathcal F\)-hypercentrality, belongs to the class \(\mathcal F\). For instance Theorem 1 states that, in this situation, the following are equivalent: (i) G is in \(\mathcal F\) (ii) Every element of order 4 of \(F(H)\) is c-normal in \(G\) and \(\langle x\rangle\) lies in the \(\mathcal F\)-hypercenter of \(G\) for every element \(x\) of prime order of \(F(H)\).
scientific article; zbMATH DE number 1787454