On some aspects of the Kegel–Wielandt $\sigma$-problem

Abstract

Let $\sigma = \{\sigma _i | i \in I \}$ be a partition of the set of all primes. A subgroup H of a finite group G is called $\sigma$-subnormal in G if there exists a chain of subgroups

$$\begin{aligned} H = H_0 \subseteq H_1 \subseteq \cdots \subseteq H_n = G \end{aligned}$$

such that for every $j = 1, 2,\ldots , n$ either $H_{j-1}$ is normal in $H_j$ or $H_j / Core_{H_j}(H_{j-1})$ is a $\sigma _i$-subgroup for some $i \in I$. Let $\pi$ be a set of primes and $G \in E_{\pi }$. A subgroup H of G is said to be $\pi$-subnormal in G if $H \cap S$ is a Hall $\pi$-subgroup of H for any Hall $\pi$-subgroup S of G. In this paper, the class of all finite groups in which all $\sigma _i$-subnormal subgroups form a join-semilattice, for any $i \in I$, is characterized. Furthermore, for such groups, the authors investigate the following Kegel–Wielandt $\sigma$-problem: Is it true that a subgroup H is $\sigma$-subnormal in G if H is $\sigma _i$-subnormal in G for all $i \in I$?