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I'm trying to answer a question that asks me to regard $S^3$ as the set of all quaternions of modulus $1$. What does this actually mean?

Thanks

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    A quaternion $a + bi + cj + dk$ has norm $a^2 + b^2 + c^2 + d^2$. The quaternions of norm $1$ can therefore be identified with $S^3$.2012-05-24
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    @QiaochuYuan Ok, thanks. How would I see that there's a subgroup of the group of homeomorphisms of $S^3$ isomorphic to $Q_8$? I can see an obvious action of $Q_8$ on $S^3$ using this identification, but how do I show the action is continuous?2012-05-24
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    Multiplication by any quaternion is continuous; in fact it can be described by polynomial functions, which are clearly continuous.2012-05-24

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