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Register a new userClassification of subgroups of symplectic groups over finite fields containing a transvection
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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.
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We describe examples motivated by the work of Serre and Abhyankar.
In previous work, the authors confirmed the speculation of J. G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups $A_n$. Here we address the natural generalization of this speculation to the finite general linear groups $mathrm{GL}_mleft(mathbb{F}_qright)$ and $mathrm{SL}_2left(mathbb{F}_qright)$.
Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as Galois groups.
Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Gamma$ determines the group $G$ upto isogeny over the algebraic closure of $K$.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $mathrm{F}_4$, where we observe that the parametrization is uniform over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
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