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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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