Binary tetrahedral group

The binary tetrahedral group $2T$ is the following subgroup of the multiplicative group of real quaternions:

2T=\{\pm1,\pm i,\pm j,\pm k\} \cup \left\{\frac{\varepsilon_0+\varepsilon_1i+\varepsilon_2j+\varepsilon_3k}{2}:\varepsilon_r\in\{1,-1\}\right\}.

All sixteen sign choices occur in the second set. These twenty-four elements have norm $1$ , and their multiplication is quaternion multiplication.

Structure

Its first eight elements form the normal quaternion subgroup $Q_8$ . The element $(-1+i+j+k)/2$ has order $3$ and conjugates the three quaternion axes cyclically, giving $2T\cong Q_8\rtimes C_3$ .

Conjugation on pure quaternions realizes

2T/\{\pm1\}\cong A_4,

the rotational tetrahedral group. Also $2T\cong\operatorname{SL}_2(\mathbb F_3)$ . Thus the binary group has order $24$ , while its rotation quotient has order $12$ .

Arithmetic realization

This is the full unit group of the Hurwitz quaternion order. The half-integral quaternions are essential: the Lipschitz order has only the eight units in $Q_8$ .

References

John Voight, Quaternion Algebras, §11.2, Lemma 11.2.1 and §§11.2.2–11.2.4.