Classification of subgroups of symplectic groups over finite fields containing a transvection

TL;DR

This paper provides a classification of subgroups of symplectic groups over finite fields, identifying conditions under which they are reducible, imprimitive, or contain the entire symplectic group, aiding in Galois representation analysis.

Contribution

It offers a self-contained proof of subgroup classification in symplectic groups over finite fields, extending Kantor's work for characteristic l ≥ 5.

Findings

Subgroups are either reducible, imprimitive, or contain Sp(n, l)

The classification is useful for big image results in Galois representations

The proof is self-contained and based on existing work

Abstract

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations.