### Abstract

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian p-groups with an abelian subgroup of index p. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete p-toral groups with an abelian subgroup of index p. In all of these cases (finite or infinite), we reduce the problem to one of listing all pG-modules (for G finite) satisfying certain conditions: a problem which was solved in the earlier paper [15] using the classification of finite simple groups.

Original language | English |
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Article number | 0308210518001075 |

Pages (from-to) | 1187-1239 |

Number of pages | 53 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 150 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jun 2020 |

### Keywords

- finite groups
- finite simple groups
- fusion
- modular representations
- Sylow subgroups
- COMPACT-GROUPS
- LIE-GROUPS
- FINITE-GROUPS
- CLASSIFICATION
- HOMOTOPY-THEORY

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## Cite this

Oliver, B., & Ruiz, A. (2020). Reduced fusion systems over p -groups with abelian subgroup of index p: III.

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*,*150*(3), 1187-1239. [0308210518001075]. https://doi.org/10.1017/prm.2018.107