On the Kegel-Wielandt \(\sigma \)-problem (Q6194909)

Let \(G\) be a finite group and let \(\sigma=\{\sigma_{i}\}_{i \in I}\) be a partition of the set of prime numbers. A subgroup \(H\) of \(G\) is called \(\sigma\)-subnormal if there exists a chain of subgroups \(H= H_{0} \leq H_{1} \leq \ldots \leq H_{n}=G\) such that, for each \(j = 1, 2, \ldots, n\), either \(H_{j-1}\) is normal in \(H_{j}\) or \(H_{j}/\mathrm{Core}_{H_{j}}(H_{j-1})\) is a \(\sigma_{i}\)-group for some \(i \in I\). The subgroup \(H\) is subnormal in \(G\) if and only if it is \(\sigma\)-subnormal in \(G\) for the minimal partition \(\sigma\) (that is \(|\sigma_{i}|=1\) for all \(i \in I\)). A system \(\Sigma=\{S_{1}, S_{2}, \ldots , S_{k} \}\) of \(\sigma_{i}\)-Hall subgroups \((i=1,2, \ldots, k\)) of \(G\) is called a complete Hall set of type \(\sigma\) of \(G\) if \(\pi(G)=\bigcup_{i=1}^{k} \sigma_{i}\). A complete Hall set \(\Sigma=\{S_{1}, S_{2}, \ldots , S_{k} \}\) of type \(\sigma\) of \(G\) reduces to a subgroup \(H \leq G\) if \(H\cap S_{i}\) is a \(\sigma_{i}\)-Hall subgroup of \(H\) for any \(i = 1, 2,\ldots,k\). Let \(\Sigma\) be a complete Hall set of type \(\sigma\) of \(G\) such that \(\Sigma^{g}\) reduces to \(H\) for any \(g \in G\). The generalized Kegel-Wielandt \(\sigma\)-problem asks if, in this case, \(H\) is \(\sigma\)-subnormal in \(G\). The main result in the paper under review is Theorem 1: Let \(\sigma\) be some partition of the set of all primes, and let \(G\) be a \(\sigma\)-complete group all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1. If \(\Sigma\) is a complete Hall set of type \(\sigma\) of \(G\), then a subgroup \(H\) of \(G\) is \(\sigma\)-subnormal in \(G\) if and only if \(\Sigma^{g}\) reduces to \(H\) for any \(g \in G\).