Permutation Groups and Point Group Symmetries

A permutation is a rearrangement of a finite number of objects. For example, one permutation of the six letters a, b, c, d, e,f is to change a into b, b into c, c into a, to transpose d and e, and to leave/alone there are several ways of denoting this permutation. One way is to list the letters twice, once in the initial, natural order and again in the revised order  

A second way is in terms of cycles in each cycle the first element is replaced by the second, the second by the third, etc., and the last by the first. Each cycle is enclosed in brackets, so that the above permutation is denoted by  

A third way of denoting permutations is in terms of matrices, where in an obvious notation the above permutation can be described as  

A permutation matrix contains only one non-zero element, a 1 in each row and column. The determinant of the matrix is then 1 if positive the permutation is called an even permutation, if negative an odd permutation. To avoid multiplying out, the parity of the permutation can be ascertained simply by counting the number of transpositions or the number of cycles of even period if this number is even, then the permutation is even if it is odd, then the permutation is odd. The number of cycles of odd period is irrelevant for determining the parity. 

In the triiodide example we considered the rearrangements of 2 and 3 elements, the atomic labels, corresponding to the permutation groups IP2 of order 2 and P3 of order 6. Readers should convince themselves that the permutations of the four labels associated with the vertices of a tetrahedron (or the ligands X of a tetrahedral MX4 molecule) form the permutation group P4 of order 24. 

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