## Abstract

This paper studies, for p a prime, rank one isolated p-minimal subgroups P in a finite group G. Such subgroups share many of the features of the minimal parabolic subgroups in groups of Lie type. The structure of Y, the normal closure in G of O ^{p}(P) is determined where O ^{p}(P) is the smallest normal subgroup of P such that P/O ^{p}(P) is a p-group. We find that if Y≠O ^{p}(P) and O _{p}(G)=1, then either Y/Z(Y) is a simple group of Lie type in characteristic p or p≤7 with Y/Z(Y) given by an explicit list. Of particular note is that twenty four out of the twenty six sporadic simple groups arise as possibilities for Y/Z(Y). This may be viewed as giving an overarching framework which brings together the simple groups of Lie type and (most of) the sporadic simple groups.

Original language | English |
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Pages (from-to) | 1-93 |

Number of pages | 93 |

Journal | Journal of Algebra |

Volume | 566 |

Early online date | 8 Sept 2020 |

DOIs | |

Publication status | Published - 15 Jan 2021 |

## Keywords

- Finite groups
- Finite simple groups
- p-minimal subgroups

## ASJC Scopus subject areas

- Algebra and Number Theory