# Proving the direct product $S^3 \times \mathbb R^{+}$ is isomorphic to $H^{*}$

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

• What are your ideas so far? And what group structure do you have on $S^3$? – John Hughes Nov 18 '15 at 18:57
• Well I'm thinking about considering the unit-1 sphere and the positive reals as that is a similar structure. Then to express this in polar form, however I am struggling with putting S3×R+ into polar form and then knowing what to do to firstly prove the homomorphism. I believe I will be okay proving injectivity and surjectivity after this. I also know that S3 is a subgroup of H* and S3 is not abelian – Jake Smith Nov 18 '15 at 19:03

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:
$$f:\mathbb{H}^*\rightarrow S^3\times\mathbb{R}^+$$ $$f(x) = \left(\frac{x}{\lVert x \rVert},\lVert x \rVert\right)$$