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**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).

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 |