Algebraic groups over finite fields: Connections between subgroups and isogenies

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_qn)n ≥ 1. We show that an isogeny Φ G' → G over F_q gives rise to a subgroup of fixed index in G(F_qn) 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_qn) diverge to infinity. Similar results are investigated regarding the sequence (G(F_p))_p by varying the characteristic p.