Let $S L(2, \mathbb{R})$ be the group of $2 \times 2$ real matrices with determinant 1 and let $\sigma: \mathbb{R} \rightarrow S L(2, \mathbb{R})$ be a homomorphism. On $K=\mathbb{R} \times \mathbb{R}^{2}$ consider the product

$$(x, \mathbf{v}) *(y, \mathbf{w})=(x+y, \mathbf{v}+\sigma(x) \mathbf{w})$$

Show that $K$ with this product is a group.

Find the homomorphism or homomorphisms $\sigma$ for which $K$ is a commutative group.

Show that the homomorphisms $\sigma$ for which the elements of the form $(0, \mathbf{v})$ with $\mathbf{v}=(a, 0), a \in \mathbb{R}$, commute with every element of $K$ are precisely those such that

$$\sigma(x)=\left(\begin{array}{cc} 1 & r(x) \\ 0 & 1 \end{array}\right)$$

with $r:(\mathbb{R},+) \rightarrow(\mathbb{R},+)$ an arbitrary homomorphism.