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Zbl 1103.20009
Glasby, S.P.; Leedham-Green, C.R.; O'Brien, E.A.
Writing projective representations over subfields. (English)
[J] J. Algebra 295, No. 1, 51-61 (2006). ISSN 0021-8693

Summary: Let $G=\langle X\rangle$ be an absolutely irreducible subgroup of $\text{GL}(d,K)$, and let $F$ be a proper subfield of the finite field $K$. We present a practical algorithm to decide constructively whether or not $G$ is conjugate to a subgroup of $\text{GL}(d,F).K^\times$, where $K^\times$ denotes the centre of $\text{GL}(d,K)$. If the derived group of $G$ also acts absolutely irreducibly, then the algorithm is Las Vegas and costs $O(|X|d^3+d^2\log|F|)$ arithmetic operations in $K$. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of $\text{GL}(d,K)$.

MSC 2000:: *20C40 Computational methods (representations of groups)
20G40 Linear algebraic groups over finite fields
20C25 Projective representations and multipliers of groups
20C33 Representations of finite groups of Lie type
20G05 Representation theory of linear algebraic groups

Keywords: absolutely irreducible subgroups; general linear groups over finite fields; algorithms; Aschbacher classes

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