## Abstract

Let x 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 x-subgroup if there exists an isomorphic embedding ϕ : G ↪ G
^{*} of G into some finite group G
^{*} under which G
^{ϕ} is subnormal in G
^{*} and H
^{ϕ} = K ∩ G
^{ϕ} for some maximal x-subgroup K of G
^{*}. In the case where x coincides with the class of all π-groups for some set π of prime numbers, submaximal x-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal x- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal x-subgroups are conjugate in a finite group G in which all maximal x-subgroups are conjugate?

Original language | English |
---|---|

Pages (from-to) | 9-28 |

Number of pages | 20 |

Journal | Algebra and Logic |

Volume | 57 |

Issue number | 1 |

DOIs | |

Publication status | Published - 19 May 2018 |

## Keywords

- Dπ-property
- finite group
- Hall π-subgroup
- maximal ð-subgroup
- submaximal ð-subgroup
- maximal x subgroup
- submaximal x-subgroup
- D-pi-pproperty
- maximal x-subgroup
- CLASSIFICATION
- Hall pi-subgroup
- CONJECTURE
- PRIMITIVE PERMUTATION-GROUPS
- FINITE-GROUPS
- HALL SUBGROUPS
- PI
- ODD INDEX
- PRONORMALITY