## Abstract

Let G be a linear algebraic group defined over a finite field Fq. We present several connections between the isogenies of G and the finite groups of rational points (G(F_{q}n)n ≥ 1. We show that an isogeny Φ G' → G over F_{q} gives rise to a subgroup of fixed index in G(F_{q}n) for infinitely many n. Conversely, we show that if G is reductive, the existence of a subgroup H_{n} of fixed index k for infinitely many n implies the existence of an isogeny of order k. In particular, we show that the infinite sequence H_{n} is covered by a finite number of isogenies. This result applies to classical groups GL_{m}, SL_{m}, SO_{m}, SU_{m}, Sp_{2m} and can be extended to non-reductive groups if k is prime to the characteristic. As a special case, we see that if G is simply connected, the minimal indices of proper subgroups of G(F_{q}n) diverge to infinity. Similar results are investigated regarding the sequence (G(F_{p}))_{p} by varying the characteristic p.

Original language | English |
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Pages (from-to) | 1143-1155 |

Number of pages | 13 |

Journal | Journal of Group Theory |

Volume | 26 |

Issue number | 6 |

Early online date | 25 Mar 2023 |

DOIs | |

Publication status | Published - 1 Nov 2023 |

### Bibliographical note

Publisher Copyright:© 2023 the author(s), published by De Gruyter.