Finite symmetry group/Tetrahedron from numerical condition/Fact/Proof

Proof

Due to the condition, there are three classes of semiaxes of order ${}2,3$ and ${}3$; the number of elements in these classes is ${}6,4,$ and ${}4$. We consider a class of semiaxes ${}K$ of order ${}3$, with the four equivalent semiaxes, and the corresponding group homomorphism

$G\longrightarrow \operatorname {Perm} \,(K),g\longmapsto \sigma _{g}.$

Let ${}g\in G$ be a rotation by ${}120$ degree around a semiaxis ${}H\in K$. It fixes ${}H$, and it gives a permutation of the three other semiaxes in the class. This permutation can not be the identity; otherwise, ${}g$ would fix two axes and then ${}g$ were the identity. Since ${}g$ has order ${}3$, this permutation is a ${}3$-cycle. In particular, the four semiaxes belong to different axes, and the rotation ${}g^{2}$ induces the other ${}3$-cycle. Since we can perform this consideration with every semiaxis from ${}K$, we can deduce that ${}G$ induces every ${}3$-cycle of the permutation group of the four semiaxes. This means that the image of the group homomorphism is exactly the alternating group ${}A_{4}$; hence, ${}G\cong A_{4}$. This group is isomorphic to the tetrahedral group, see exercise.

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