General linear group over integers

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Definition

Let n be a natural number. The general linear group of order n over integers, denoted GL(n,\mathbb{Z}) or GL_n(\mathbb{Z}), is defined in the following equivalent ways:

  1. It is the general linear group of order n over the ring \mathbb{Z}, defined as the ring of integers.
  2. It is the automorphism group of the free abelian group of rank n, i.e., the group \mathbb{Z}^n.