Algebraic Groups over Finite Fields: Connections Between Subgroups and Isogenies

Let G be a linear algebraic group defined over a finite field F_q. We present several connections between the isogenies of G and the finite groups of rational points G(F_q^n). We show that an isogeny from G' to 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 of fixed index k for infinitely many n implies the existence of an isogeny of order k. In particular, we show that every infinite sequence of subgroups is controlled by a finite number of isogenies. This result applies to classical groups GLm, SLm, SOm, SUm, Sp2m 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 indexes of proper subgroups of G(F_q^n) diverge to infinity. Similar results are investigated regarding the sequence G(F_p) by varying the characteristic p.