My question refers to following former question of mine: General Linear Group $GL_n(R)$ not Finitely Generated
I want to know how to see that the general linear group $GL_n(\mathbb{Z})$ of integers is finitely generated.
2026-02-23 00:28:06.1771806486
General Linear Group $GL_n(\mathbb{Z})$ of Integers is finitely generated
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The Smith normal form algorithm shows that a matrix in $GL_n(\mathbb{Z})$ can be diagonalized by elementary row and column operations. A row operation is the same as multiplying on the left by a power of an elementary matrix (a matrix with all $1$'s on the diagonal and a single off diagonal $1$). A column operation is the same as multiplying on the right by a power of an elementary matrix. A non-identity diagonal matrix in $GL_n(\mathbb{Z})$ is a product of special diagonal matrices having all $1$'s except for a single $-1$ on the diagonal. So $GL_n(\mathbb{Z})$ is generated by elementary matrices and special diagonal matrices.