Consider the direct product of the unit 3-sphere with the positive real numbers:
$S^3 \times \mathbb R^{+}$
Prove that this group is isomorphic to the non-zero quaternions $H^{*}$ under multiplication.
Thanks
2025-01-13 05:46:17.1736747177
Proving the direct product $S^3 \times \mathbb R^{+}$ is isomorphic to $H^{*}$
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The group of unit quaternions is isomorphic to $S^3$. So, decomposing a quaternion into its norm and a unit quaternion provides the required isomorphism:
$$f:\mathbb{H}^*\rightarrow S^3\times\mathbb{R}^+$$ $$f(x) = \left(\frac{x}{\lVert x \rVert},\lVert x \rVert\right)$$