Hi everyone,

I'm working on this problem of proving that two groups are isomorphic. The two groups in question are the full tetrahedral group $\displaystyle T_h$ of all permutations on the tetrahedron (isomorphic to $\displaystyle S_4$) and the group which I've named $\displaystyle G_6j$, which is the group of symmetries on the 6j symbol.

Definition and symmetries of the 6j symbol can be found here.

6-j symbol - Wikipedia, the free encyclopedia

Basically, a 6j symbol is an 2x3 array with indices $\displaystyle a_1,a_2,a_3,...,a_6$ which is invariant under any permutation of the columns and the interchanging of upper and lower- indices in pairs of columns. This yields a total of 24 symmetry operations on the 6j symbol. Furthermore one can visualize such a symbol as a (regular) tetrahedron with sides $\displaystyle a_1,a_2,...,a_6$, as is depicted in DLMF: §34.4 Definition: 6j6j Symbol .

Now I wish to show that these two groups are isomorphic. I see that the symmetry operations of the first kind correspond to permuting a face of the tetradron (this is essentially $\displaystyle S_3$), and that the symmetry operations of the second kind correspond to rotating the tetrahedron, i.e. swichting faces. So, this way, one can (bijectively) generate all permutations of the tetrahedron with all symmetry operations on the 6j symbol.

I hope this is correct so far. I would then have proven a bijection between the two groups. How would I go on and prove the homomorphic property of this map?

Thanks