A general linear group over a field is finite if and only if the field is finite

Let F be a field. Show that GL_n(F) is a finite group if and only if F is finite.

Suppose F is finite. Then there are only finitely many n \times n matrices over F, in particular, |F|^{n^2}. Thus there are at most |F|^{n^2} elements in GL_n(F).

Suppose now that F is infinite. Note that for all \alpha \in F, the matrix A_\alpha defined such that a_{i,j} = \alpha if i = j = 1, 1 if i = j \neq 1, and 0 otherwise has determinant \alpha and so is in GL_n(F). Thus GL_n(F) is infinite.

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