Unit quaternion
= Unit quaternion
{title2=$\operatorname{Sp}(1)\cong S^3$}
A <quaternion> of norm $1$ is a unit quaternion. They form the <topological group> $\operatorname{Sp}(1)$ and identify with the sphere $S^3\subset\mathbb R^4$. Multiplication and inversion preserve the norm. The <Hurewicz theorem> identifies $\pi_3(S^3)$ with $H_3(S^3;\mathbb Z)$; hence <homotopy-group addition in a topological group> shows that $q\mapsto q^m$ has <mapping degree> $m$, rather than $m^2$.