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Exercise :

Consider a regular tetrahedron with its vertices named $1,2,3,4$. Describe geometrically the symmetries that correlate to the permutations $(12),(123)$ and $(1234)$. Furthermore, show that the group of permutations $S_4$ the group of symmetries of the tetrahedron $\Sigma _4$ are isomorphic and that the group $A_4$ correlates to the rotation subgroup $R_4$.

Attempt (more like I complete question) :

So, I fetched up a sketch and named each vertex with the corresponding number :

enter image description here

Now, if we take the axis that passes through the mid-points of the segment $1-2$ and $3-4$ and rotate it by $2\pi /3$, I guess that $(12)(34)$ is an even permutation. So, if I'm correct, the even permutation $(12)(34)$ in this case, is represented geometrically as the rotation of a regular tetrahedron by the axis elaborated by $2 \pi /3$.

My question is though, how would I need to proceed in order to describe geometrically all the symmetries that are subject to the permutations named ?

For the second part, starting from proving the isomorphic fact between $S_4$ and $\Sigma _4$, we already have a regular tetrahedron with its vertices numbered. Every element of the full tetrahedral group permutes the vertices of the regular tetrahedron among themselves. Moreover, composing elements of the group composes the actions on the vertices. Therefore, the full tetrahedral group has a group action on the set $\{1,2,3,4\}$. Therefore, there is a homomorphism of groups from the full tetrahedral group to symmetric group $S_4$ defined by this action.

Is the above elaboration enough to answer the first question of the second part ?

Question : How would I show after these, that the even permutations group $A_4$ corresponds to the subgroup of rotations $R_4$ ?

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  • $\begingroup$ A rotation through an angle $2\pi/3$ is of order three (its third power is the identity). Is that the case for the permutation $(1\ 2)(3\ 4)$? $\endgroup$ – Lord Shark the Unknown Jan 20 '18 at 21:14

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