Exercise 1.4.6 ($\mathrm{GL}_n(F)

Question

If $|F| = q$ is finite prove that $|\mathrm{GL}_n(F)| < q^{n^2}$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $\mathscr{M}_n(F)$ be the set of $n 	imes n$ matrices. The null matrix is not invertible, so
$$\mathrm{GL}_n(F) \subset \mathscr{M}_n(F) \setminus \{0\}.$$
The cardinality of $\mathscr{M}_n(F)$ is $q^{n^2}$, since the $n^2$ entries are in $F$ of cardinality $q$.
Therefore
$$|\mathrm{GL}_n(F)| < q^{n^2}.$$

\end{proof}

Exercise 1.4.6 ($\mathrm{GL}_n(F) < q^{n^2}$)

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Exercise 1.4.6 ($\mathrm{GL}_n(F) < q^{n^2}$)

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