Abstract
By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).
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The second author wishes to thank Tim Penttila and Mary Worthley for their kind hospitality and for many stimulating conversations.
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Penttila, T., Siciliano, A. On collineation groups of finite projective spaces containing a Singer cycle. J. Geom. 107, 617–626 (2016). https://doi.org/10.1007/s00022-015-0300-4
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DOI: https://doi.org/10.1007/s00022-015-0300-4