The order of a general linear group over a finite field is bounded above

Let F be a field. If |F| = q is finite show that |GL_n(F)| < q^{n^2}.


Clearly GL_n(F) is contained in the set of all n \times n matrices over F. There are at most q^{n^2} such matrices, so |GL_n(F)| \leq q^{n^2}. Moreover, the zero matrix is not in GL_n(F), so the inequality is strict.

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