Exercise 3.5.7 (The group of the rigid motions of a tetrahedron is isomorphic to $A_4$)

Question

Prove that the group of the rigid motions of a tetrahedron is isomorphic to $A_4$. [Recall Exercise 20 in Section 1.7.]

Richard Ganaye · Accepted Answer

\begin{proof} 
We have proved in Exercise 1.7.20 that
$$
\varphi
\left\{
\begin{array}{ccl}
G & 	o &S_T\
g & \mapsto &\varphi_g = \begin{pmatrix} A & B & C & D\g(A)& g(B) & g(C) & g(D) \end{pmatrix}.
\end{array}
ight.
$$
is an injective homomorphism, which maps the group $G$ of rigid motions of a regular tetrahedron on a subgroup of $S_T \simeq S_4$.


By Exercise 1.2.9, (see Proposition 4 of the solution), the isometry $f$ defined by $A_i \mapsto A_{	au(i)}$ is  a rigid motion if and only if $	au \in A_4$. Therefore
the group of the rigid motions of a regular tetrahedron is isomorphic to $A_4$.

\end{proof}

Exercise 3.5.7 (The group of the rigid motions of a tetrahedron is isomorphic to $A_4$)

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Exercise 3.5.7 (The group of the rigid motions of a tetrahedron is isomorphic to $A_4$)

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