The question is

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> Describe the elements of the group of proper symmetries of a regular tetrahedron as a subgroup of $S\_4$. Show that -- besides the identity -- there are two sorts of rotations -- one fixing only one vertex and the other fixing no vertex.
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I don't understand what the elements are referring to.

I've seen in different places that $S\_4$ is supposed to be the symmetric group on $n$ elements, but my textbook says "The symmetry groups of the Platonic solids are quite interesting finite groups called the tetrahedral group ($A\_4$) of proper symmetries of a tetrahedron, the octahedral group ($S\_4$) of proper symmetries of an octahedron or cube"? Honestly, I might just be completely lost on this. The first question "Describe the elements of the group of proper symmetries of a regular tetrahedron as a subgroup of $S\_4$" makes no sense to me.

Are the symmetries on the tetrahedron the elements of the group of proper symmetries? Is that how it works?