On saturated formations, theta-pairs and completions in finite groups
The author uses formation theory to obtain generalizations and improvements of results of \textit{N. P. Mukherjee} and \textit{P. Bhattacharya} [Proc. Am. Math. Soc. 109, No. 3, 589-596 (1990; Zbl 0699.20019)] and \textit{J. C. Beidleman} and \textit{H. Smith} [ibid. 114, No. 2, 319-324 (1992; Zbl 0753.20011) and Publ. Math. (to appear)] for finite groups. For example, let \(F\) be a saturated formation and \(G\) be a finite group having a maximal subgroup \(M\) in \(F\). Suppose further that \((\text{Fit}(G),M\cap\text{Fit}(G))\) is a \(\theta\)-pair for \(M\) in \(G\) such that \(\text{Fit}(G)/M\cap\text{Fit}(G)\) is \(F\)-central in \(G\). Then \(G\in F\). (Theorem 4.) This generalizes Theorem 1 of the second paper of Beidleman and Smith for finite groups. Note: Let \(f\) be the canonical local definition of saturated formation and let \(B\leq A\) be subgroups of group \(G\), \(B\triangleleft G\). If the \(f(p)\) residual subgroup of \(G/B\) centralizes \(A/B\) for each prime \(p\) dividing the order of \(A/B\), then \(A/B\) is called \(F\)-central in \(G\).
scientific article; zbMATH DE number 1023959