Families of solvable Frobenius subgroups in finite groups

Abstract

We introduce the notion of abelian system on a finite group $G$, as a particular case of the recently defined notion of kernel system (see this Journal, September 2001). Using a famous result of Suzuki on CN-groups, we determine all finite groups with abelian systems. Except for some degenerate cases, they turn out to be special linear group of rank $2$ over fields of characteristic $2$ or Suzuki groups. Our ideas were heavily influenced by [1] and [8].