Group theory and generalizations Theorem

Order of the general linear group over a finite field

Statement

Let $q$ be a prime power, let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $n \geq 1$ . The group $\mathrm{GL}_n(\mathbb{F}_q)$ of the invertible $n \times n$ matrices over $\mathbb{F}_q$ has order $\prod_{k=0}^{n-1} (q^n - q^k) = (q^n - 1)(q^n - q) \cdots (q^n - q^{n-1})$ .

Remarks

A matrix is invertible exactly when its columns are linearly independent. There are $q^n - 1$ choices for the first column. Column $k+1$ has $q^n - q^k$ choices, because it must avoid the $q^k$ vectors in the span of the first $k$ columns. For example, $|\mathrm{GL}_2(\mathbb{F}_2)| = 3 \cdot 2 = 6$ , and $\mathrm{GL}_2(\mathbb{F}_2) \cong S_3$ . Also, $|\mathrm{GL}_3(\mathbb{F}_2)| = 7 \cdot 6 \cdot 4 = 168$ .

History

Évariste Galois computed the order of the general linear group over a prime field in 1832, in his last letter to Chevalier.

Reading path

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Group concept
Order of a group definition
Order of the general linear group over a finite field statement

Tag	`0JQ1KNQW2K`
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Order of the general linear group over a finite field

Statement

Remarks

History

Reading path

Further reading

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