# Group of rototional symmetires

1. Oct 17, 2006

### Nusc

Would anyone please tell me the group of rot. symm. of a regular tetrahedron?

Thanks

2. Oct 17, 2006

### matt grime

It is what it is: the group of all rigid rotations of the tetrahedron. Perhaps you want some nice description of it in terms of generators and relations? Or a group you're happy with in some sense and an isomorphism to it? Your question is highly subjective in the Clintonian 'depends on what the meaning of is is' sense. You can label the 4 vertices of the tetrahedron and just write down the group as a subgroup of S_4 by hand: it has 12 elements as can be seen by just considerin what happens to vertices. One vertex is mapped to any of four, and a neigbouring vertex is mapped to one of the three remaning ones, since you must preserve orientation this fixes the symmetry and there are 12 elements of the group.

It is therefore easy to find a set of elements of S_4 that must generate a copy of the symmetry group: write down some obvious elements, such as rotations that fix one of the vertices. how many elements is this? Now apply some of the results you know about groups.