# Spreads and classes of maximal subgroups of*GL*_{ n }*(q)*,*SL*_{ n }*(q)*,*PGL*_{ n }*(q)* and*PSL*_{ n }*(q)*

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## Summary

If r divides n then the points of PG(n−1, q) can be partitioned by the (r−1)-subspaees of a classical spread S_{r}. The underlying finite geometry of this configuration, in particular the orbits of lines, is used to prove that if r is a proper prime divisor of n then the stabilizers of S_{r} in PGL_{n}(g) and PSL_{n}(q) are maximal subgroups of PGL_{n}(q) and PSL_{n}(q respectively. Special attention is needed for the case of PSL_{n}(q) when n/r=2 and r divides q− 1. An explicit description is found for the stablizers.

## Keywords

Maximal Subgroup Prime Divisor Explicit Description Finite Geometry Classical Spread
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© Fondazione Annali di Matematica Pura ed Applicata 1991