Algebra and Geometry | Part IA, 2006

Let SL(2,R)S L(2, \mathbb{R}) be the group of 2×22 \times 2 real matrices with determinant 1 and let σ:RSL(2,R)\sigma: \mathbb{R} \rightarrow S L(2, \mathbb{R}) be a homomorphism. On K=R×R2K=\mathbb{R} \times \mathbb{R}^{2} consider the product

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

Show that KK with this product is a group.

Find the homomorphism or homomorphisms σ\sigma for which KK is a commutative group.

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

σ(x)=(1r(x)01)\sigma(x)=\left(\begin{array}{cc} 1 & r(x) \\ 0 & 1 \end{array}\right)

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

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