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Found problems: 1

2006 Cezar Ivănescu, 2
Tags: group theory , abstract algebra , matrices , isomorphism , group morphism
Prove that the set $ \left\{ \left. \begin{pmatrix} \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2} & \frac{1+x^3}{3x^2} \\ \frac{1+x^3}{3x^2} & \frac{1+x^3}{3x^2} & \frac{1-2x^3}{3x^2}\end{pmatrix}\right| x\in\mathbb{R}^{*} \right\} $ along with the usual multiplication of matrices form a group, determine an isomorphism between this group and the group of multiplicative real numbers.