Unit quaternion

2020 Mathematics Subject Classification: Primary: 11R52 [MSN][ZBL]

A quaternion with norm 1, that is, $xi+yj+zk+t$ with $x^2+y^2+z^2+t^2=1$.

The real unit quaternions form a group isomorphic to the special unitary group ${\mathrm{SU}}_2$ over the complex numbers, and to the spin group ${\mathrm{Sp}}_3$. They double cover the rotation group ${\mathrm{SO}}_3$ with kernel $\pm 1$ (cf. rotations diagram).

The finite subgroups of the unit quaternions are given by group presentations $$A^p=B^q=(AB{)}^2$$ with $1/p+1/q>1/2$, denoted $⟨p,q,2⟩$. They are

• the cyclic groups $C_n$, , corresponding to $⟨n,n,1⟩$;
• the dicyclic groups, corresponding to $⟨n,2,2⟩$;
• the binary tetrahedral group $⟨3,3,2⟩$;
• the binary octahedral group $⟨4,3,2⟩$;
• the binary icosahedral group $⟨5,3,2⟩$.

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