Prove that the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated [duplicate]

linear-algebra abstract-algebra group-theory finitely-generated

Solution 1:

It is known, that for any Euclidean ring $R$ that $GL_n(R)$ is generated by the elementary matrices (proved via Gaussian elimination). For $\mathbb{Z}$ there are finitely many of them (as there are finitely many invertible elements)

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