Algebraic groups over finite fields: Connections between subgroups and isogenies

Abstract Let 𝐺 be a linear algebraic group defined over a finite field F q \mathbb{F}_{q} . We present several connections between the isogenies of 𝐺 and the finite groups of rational points ( G ⁢ ( F q n ) ) n ≥ 1 (G(\mathbb{F}_{\smash{q^{n}}}))_{n\geq 1} . We show that an isogeny ϕ : G ′ → G \phi\colon G^{\prime}\to G over F q \mathbb{F}_{q} gives rise to a subgroup of fixed index in G ⁢ ( F q n ) G(\mathbb{F}_{\smash{q^{n}}}) for infinitely many 𝑛. Conversely, we show that if 𝐺 is reductive, the existence of a subgroup H n 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 H_{n} is covered by a finite number of isogenies. This result applies to classical groups GL m \mathrm{GL}_{m} , SL m \mathrm{SL}_{m} , SO m \mathrm{SO}_{m} , SU m \mathrm{SU}_{m} , Sp 2 ⁢ m \mathrm{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 ) G(\mathbb{F}_{\smash{q^{n}}}) diverge to infinity. Similar results are investigated regarding the sequence ( G ⁢ ( F p ) ) p (G(\mathbb{F}_{p}))_{p} by varying the characteristic 𝑝.