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Last edited by hbghlyj at 2023-2-3 14:21:00$$|\mathrm{GL}\left(n, q\right)|=\left(q^{n}-1\right)\left(q^{n}-q\right) \cdots\left(q^{n}-q^{n-1}\right)
$$Proof. $|\mathrm{GL}(n,q)|$ is the number of $n×n$ matrices in $𝔽_q$ whose rows are linearly independent.
To construct such a matrix, we can choose any non-zero vector in $𝔽_q^n$ as the first row; there are $q^n-1$ such choices.
For $1<k\le n$, the $k$th row can be any vector in $𝔽_q^n$ except for the $q^{k-1}$ linear combinations of the previous $k-1$ rows; hence there are $q^n-q^{k-1}$ choices for the $k$th row.
The stated formula now follows. $_\blacksquare$ |
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青青子衿
Posted at 2023-2-3 10:17:29
Czhang271828
Posted at 2023-2-3 13:10:52
