## Abstract

Let ν be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal ν-subgroup if there exists an isomorpic embedding Ο: G βͺ G^{*} of the group G into some finite group G^{*} under which G^{Ο} is subnormal in G^{*} and H^{Ο} = K β©G^{Ο} for some maximal ν-subgroup K of G^{*}. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal ν-subgroups are conjugate in a finite group G in which all maximal ν-subgroups are conjugate? This question strengthens Wielandtβs known problem of closedness for the class of [InlineMediaObject not available: see fulltext.]-groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group.

Original language | English |
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Pages (from-to) | 169-181 |

Number of pages | 13 |

Journal | Algebra and Logic |

Volume | 57 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jul 2018 |

## Keywords

- finite group
- Hall Ο-subgroup
- maximal π-subgroup
- submaximal π-subgroup
- [InlineMediaObject not available: see fulltext.]-property
- Hall p-subgroup
- Dpproperty
- DX-property.
- maximal X-subgroup
- HALL SUBGROUPS
- FINITE-GROUPS
- submaximal X-subgroup