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Does anyone know, where I can find a reference (preferably a book) which says that the general linear group $\text{GL}_{2}(\mathbb{Z})$ is generated by the set

$$\left\{\begin{bmatrix} 1&0\\0&-1\end{bmatrix}, \begin{bmatrix} 0&1\\1&0\end{bmatrix}, \begin{bmatrix} 1&1\\0&1\end{bmatrix}\right\}$$

Thank you in advance.

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  • $\begingroup$ it is easy to prove without any book $\endgroup$
    – Leox
    Commented Oct 16, 2014 at 8:40
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    $\begingroup$ I am looking just for a reference, not a proof. I don't want to include such trivial proof in a master thesis. $\endgroup$
    – H.E
    Commented Oct 16, 2014 at 8:42
  • $\begingroup$ I am not sure but try look in S.Lang, SL(2) $\endgroup$
    – Leox
    Commented Oct 16, 2014 at 8:54

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You could cite Magnus, Karrass and Solitar's book Combinatorial group theory. If I were you I would cite Theorem 3.2 (p131 - this says that certain Nielsen transformations generate $F_2$) and Corollary N4 (p169 - this says that these Nielsen transformations also generate $GL_2(\mathbb{Z})$ in a natural way: the way you want). However, there may be other "citation paths" to this result using their book, and you may be more comfortable with one of these. Certainly - the result follows easily from the book!

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  • $\begingroup$ Thank you. I will check this source. $\endgroup$
    – H.E
    Commented Oct 16, 2014 at 9:41

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