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Let G be a linear algebraic group defined over a finite field F_q. We presentseveral connections between the isogenies of G and the finite groups ofrational points G(F_q^n). We show that an isogeny from G' to G over F_q givesrise 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 fixedindex 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 controlledby 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 primeto the characteristic. As a special case, we see that if G is simply connectedthe 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 thecharacteristic p.

Algebraic Groups over Finite Fields: Connections Between Subgroups and Isogenies