### Abstract

© 2015 by the authors. We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4.

Original language | English |
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Pages (from-to) | 7665-7694 |

Number of pages | 30 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 11 |

Early online date | 23 Mar 2015 |

Publication status | Published - Nov 2015 |

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## Cite this

Guest, S., Morris, J., Praeger, C., & Spiga, P. (2015). On the maximum orders of elements of finite almost simple groups and primitive permutation groups.

*Transactions of the American Mathematical Society*,*367*(11), 7665-7694.