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


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.6

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


Solution: 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.


Linearity

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