Algebraic Groups over Finite Fields: Connections Between Subgroups and Isogenies

TL;DR

This paper explores the relationship between isogenies of algebraic groups over finite fields and the structure of their rational point subgroups, revealing how subgroup sequences are governed by isogenies, with implications for classical groups.

Contribution

It establishes a link between isogenies and subgroup indices in finite field algebraic groups, extending results to non-reductive groups and classical groups, and analyzing subgroup behavior over varying characteristics.

Findings

Isogenies correspond to subgroups of fixed index in G(F_q^n) for infinitely many n.

Sequences of subgroups are controlled by finitely many isogenies.

Minimal indexes of proper subgroups diverge for simply connected groups.

Abstract

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.