\(D_{\pi}\)-property and normal subgroups
Let G be a finite group and \(\pi\) a set of primes. By definition, G satisfies \(D_{\pi}\) if G has a Hall \(\pi\)-subgroup H and each \(\pi\)- subgroup of G is contained in a conjugate of H. It is unknown whether the \(D_{\pi}\)-property is inherited by normal subgroups (conjecture of F. Gross). The main result of the paper under review is the following Theorem 1.4. Let G be a finite group satisfying both \(D_{\pi}\) and \(D_{\pi '}\), and \(N\triangleleft G\). Then N satisfies \(D_{\pi}\) and \(D_{\pi '}\). Here \(\pi\) ' is the set of all primes not in \(\pi\). - The following result is a corollary of Theorem 1.4 and of a theorem of Z. Arad and E. Fisman. Theorem 3.2. Let G be a finite group. It satisfies \(D_{\pi}\) and \(D_{\pi '}\) if and only if the composition factors of G are of the following types: 1) \(\pi\)-groups; 2) \(\pi\) '-groups; 3) simple groups \(PSL_ 2(q)\) for \(q>3\), q(q-1)\(\equiv 0 (mod 3)\), \(q\equiv -1 (mod 4)\), \(\pi (q+1)\subseteq \pi\) and \(\pi\) (q(q-1))\(\subseteq \pi '\).
scientific article; zbMATH DE number 4038076