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 $\langle p,q,2 \rangle$. They are

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

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