On join properties of Hall \(\pi\)-subgroups of finite \(\pi\)-soluble groups
Let \(\pi\) be a set of primes and let \(\text{Hall}_\pi(G)\) denote the set of all Hall \(\pi\)-subgroups of a finite group \(G\). \(G\) has property \(J_\pi\) (resp. \(J_\pi^*\)) if, whenever \(H\in\text{Hall}_\pi(G)\) and \(U,V\leq G\) such that \(H\cap U\in\text{Hall}_\pi(U)\) and \(H\cap V\in\text{Hall}_\pi(V)\), then \(H\cap\langle U,V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\) (resp. \(\langle H\cap U,H\cap V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\)) holds. The authors among others show that these properties are equivalent, if \(G\) is \(\pi\)-soluble, to \(G/O_\pi(G)\) (resp. \(G\)) has a normal \(\pi\)-complement. Furthermore they investigate the following question: Let \(H\in\text{Hall}_\pi(G)\) and \(U,V\leq G\) be such that \(H\cap U\in\text{Hall}_\pi(U)\) and \(H\cap V\in\text{Hall}_\pi(V)\). Which conditions imply that \(\langle H\cap U,H\cap V\rangle\in\text{Hall}_\pi(\langle U,V\rangle)\)? In particular they derive an improvement of Wielandt's result on the behaviour of Hall \(\pi\)-subgroups in groups generated by subnormal subgroups.
scientific article; zbMATH DE number 1218673