Groups with many self-normalizing subgroups.(English)Zbl 1199.20065

Summary: This paper investigates the structure of groups in which all members of a given relevant set of subgroups are self-normalizing. In particular, soluble groups in which every non-Abelian (or every infinite non-Abelian) subgroup is self-normalizing are described.
Let $$\mathcal H$$ be the class of all groups in which every non-Abelian subgroup is self-normalizing. Clearly, all groups whose proper subgroups are Abelian belong to the class $$\mathcal H$$, and we shall denote by $$\mathcal H^*$$ the class of $$\mathcal H$$-groups containing proper non-Abelian subgroups; it will be proved that the structure of soluble non-nilpotent $$\mathcal H^*$$-groups is close to that of minimal non-Abelian groups. Moreover, if $$\mathcal H_\infty$$ denotes the class of groups in which all infinite non-Abelian subgroups are self-normalizing, it will turn out that any soluble $$\mathcal H_\infty$$-group either has the property $$\mathcal H$$ or is a Chernikov group (i.e. it is a finite extension of an Abelian group satisfying the minimal condition on subgroups).

MSC:

 20F19 Generalizations of solvable and nilpotent groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20E07 Subgroup theorems; subgroup growth 20F14 Derived series, central series, and generalizations for groups