The structure of intersections of nilpotent \(\pi\)-subgroups in finite \(\pi\)-solvable groups
Let \(p\) be a prime number and \(G\) be a finite group. \textit{D. Passman} proved [Trans. Am. Math. Soc. 123, 99-111 (1966; Zbl 0139.018)] that, for any Sylow \(p\)-subgroup \(P\) of \(G\), there exists a pair \(P^ x\), \(P^ y\) of conjugates of \(P\) such that \(P \cap P^ x \cap P^ y = O_ p(G)\). Here \(O_ p(G)\) is the largest normal \(p\)-subgroup of \(G\). Generally, there is no subgroup \(P^ x\) such that \(P\cap P^ x = O_ p(G)\). In the paper under review, the following result is proved. Theorem. Let \(G\) be a finite \(\pi\)-solvable group with a nilpotent Hall \(\pi\)-subgroup \(H\), and let \(H_ 1,H_ 2,\dots,H_ n\) be an arbitrary collection of Hall \(\pi\)-subgroups of \(G\). Then there is a pair \(H^ u\), \(H^ v\) of Hall \(\pi\)-subgroups of \(G\) such that \(H_ 1 \cap H_ 2 \cap \dots \cap H_ n = H^ u \cap H^ v\). (Note that all Hall \(\pi\)-subgroups in \(G\) are conjugate since \(H\) is nilpotent).
scientific article; zbMATH DE number 652177