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

De Falco, M.; de Giovanni, F.; Musella, C.

Algebra Discrete Math. 2009, No. 4, 55-65 (2009).

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

Keywords:

self-normalizing subgroups; minimal non-Abelian groups; generalized soluble groups; non-nilpotent groups; Chernikov groups; minimal condition on subgroups; classes of groups