Gl group of ut.

ALGEBRA - LECTURE II 1. General Linear Group Let F q be a finite field of order q. Then GL n(q), the general linear group over the field F q, is the group of invertible n × n matrices with coefficients in F q.We shall now compute the order of this group. Note that columns of an invertible matrix give a basis of V = Fn q.

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that the order of UT is |UT| = |U||T| |U ∩T|. Example: Let p be a prime and G = GL 3(p). Let U be the group of upper triangular matrices with 1 on the diagonal (so called unipotent matrices), and let T be the group of all diagonal matrices. Then T normalizes U. Since U ∩T = {1} the group UT is isomorphic to the semi-direct product of U and ...

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