A class of generalised finite T-groups

Let F be a formation (of finite groups) containing all nilpotent groups such that any normal subgroup of any T-group in F and any subgroup of any soluble T-group in F belongs to F. A subgroup M of a finite group G is said to be F-normal in G if G/CoreG(M) belongs to F. Named after Kegel, a subgroup U of a finite group G is called a K-F-subnormal subgroup of G if either U=G or U=U0⩽U1⩽⋯⩽Un=G such that Ui−1 is either normal in Ui or Ui−1 is F-normal in Ui, for i=1,2,…,n. We call a finite group G a TF-group if every K-F-subnormal subgroup of G is normal in G. When F is the class of all finite nilpotent groups, the TF-groups are precisely the T-groups. The aim of this paper is to analyse the structure of the TF-groups and show that in many cases TF is much more restrictive than T.