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**Problem 7.**  
Let \(\mathbb{Z}\) denote the set of general linear functions over \(\mathbb{Z}\). So for \(\psi, \phi \in \mathbb{Z}\), \(\psi\) and \(\phi\) are invertible and for every \(\psi \in \mathbb{Z}\), \(\psi^{-1}\) exists. Also let \(\mathbb{Z}^2 = \mathbb{Z} \times \mathbb{Z}\). The standard action of \(\mathbb{Z}\) on \(\mathbb{Z}^2\) is given by \(\psi \cdot (x, y) = (\psi(x), \psi(y))\). We now consider the skew-action, defined as \(\psi \cdot (x, y) = (\psi(x), \psi(y) + \alpha x)\) for some fixed \(\alpha \in \mathbb{Z}\). Prove or disprove the following:

1. \([x] = [y]\) if and only if \([x] = [z]\) for all \(x, y, z \in \mathbb{Z}^2\) where \(\psi \in \mathbb{Z}\).
   
2. There exists a unique \(\gamma: \mathbb{Z}^2 \to \mathbb{Z}\) such that \(\gamma(x, y)\) is invariant under \(\mathbb{Z}\).

3. Suppose \(n\) is a fixed nonzero integer. Prove that \(n^2 + y^2 = z^2 + n\) where \(y, z \in \mathbb{Z}\).

4. \(\mathbb{Z}\) has a unique nontrivial automorphism \(\alpha: \mathbb{Z} \to \mathbb{Z}\) which satisfies \(g \cdot (x, y) = (\phi(x), \phi(y))\) for all \(\phi, \psi \in \mathbb{Z}\).

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Dive into the complex world of group actions and automorphisms in linear algebra. Solve and explore problems involving linear functions over sets like \( \mathbb{Z} \) and \( \mathbb{Z}^2 \), including unique automorphisms and group actions. Suitable for advanced undergraduate and graduate students.

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