On weakly subnormal subgroups which are not subnormal (Q1820238)

A subgroup H of a group G is said to be n-step weakly subnormal in G (written \(H\leq ^ nG)\), for some integer \(n\geq 0\), if there are subsets \(S_ i\) of G such that \(H=S_ 0\subseteq S_ 1\subseteq...\subseteq S_ n=G\) with \(u^{-1}Hu\subseteq S_ i\) for all \(u\in S_{i+1}\), \(0\leq i\leq n-1\). Subnormal subgroups are clearly weakly subnormal and it is known that the two concepts coincide in both finite groups and soluble groups. Also \(H\leq ^ 2G\) always implies that \(H\triangleleft ^ 2G\). In the present work it is shown that a weakly subnormal subgroup is not subnormal in general. In particular (Theorem A) there is a group G with a 3-step weakly subnormal subgroup H which is not subnormal in G. Moreover, for all elements g of G, \(H\triangleleft ^ 3<H,g>\) (Theorem B). The group G is locally finite and H is finite and there is the following Corollary: Let \(n\geq 0\) be any integer. Then there is a finite group F with a subgroup H such that \(H\triangleleft ^ 3<H,f>\) for all \(f\in F\), but the subnormal defect of H in F exceeds n. (Of course H is subnormal in F by a famous result of Wielandt.)