Group of direct symmetries of a regular tetrahedron

Question

I'm trying to understand the group of direct symmetries of the regular tetrahedron ($\mathcal{T}$), $$ G_+ := \operatorname{Sym}(G) \cap \mathcal{M}_+, $$ where $\mathcal{M}_+$ is the set of rotations and translations. So in this case, I'm thinking that $G_+$ only consists on the rotations of the vertices.

1. First I'm trying to show that $G_+$ is finite. In order to do this, I was thinking on considering the action of $G_+$ on the vertices of $\mathcal{T}$, and then "counting" how many ways I can rotate the vertices (this gives 12, if I'm not mistaken, so I conclude that $G_+$ is finite).

2. Then I want to show that if $c$ is the centre of the tetrahedron, and $\varphi \in G_+$, then $\varphi(c) = c$, so $\varphi$ is a rotation whose axis contains the point $c$. In order to do this, I thought about using again the fact that the direct movements are the rotations and translations, but I can't find a way to make this work.

3. Finally I want to show that this action is transitive, prove that the stabilizer of a face is the cyclic group $C_3$. This would let me again deduce that the order of $G_+$ is 12, and describe its elements. In this final step, I honestly don't know what to consider, other than the fact that $G_+$ acts naturally on the group of faces of $\mathcal{T}$.

As always, any help is more than welcome. Thanks in advance!

EDIT: I think that instead of "this action is transitive", it would be better to say "this action acts transitively".

Answer 1

Interesting, I’ve dealt with symmetries a lot over the years, and I’d never encountered the terms “direct” and “indirect”symmetries before.

You’re indeed mistaken about your count of the ways you can rotate the vertices. For each axis through a vertex and the centre of the opposite face, there are $2$ different rotations through $\frac{2\pi}3$. I suspect you counted $3$ rotations per vertex because you included the identity; but the identity is the same for all four vertices, so it only counts once. That leaves three symmetries to be accounted for (since you’re right that there are $12$), and these are the rotations through $\pi$ about the axes connecting the midpoints of opposite edges.