Mathematical Methods for Physics and Engineering - Matematica.NET

GROUP THEORYeach number appears once and only once in the representation of any particularpermutation.The order of any permutation of degree n within the group S n canbereadofffrom the cyclic representation and is given by the lowest common multiple (LCM)of the lengths of the cycles. Thus I has order 1, as it must, and the permutationθ discussed above has order 4 (the LCM of 4 and 2).Expressed in cycle notation our second permutation φ is (3)(1 4 6)(2 5), andthe product φ • θ is calculated as(3)(1 4 6)(2 5) • (1 2 5 4)(3 6){a bcdef} = (3)(1 4 6)(2 5){b efadc}= {a dfceb}= (1)(5)(2 4 3 6){a bcdef}.i.e. expressed as a relationship amongst the elements of the group of permutationsof degree 6 (not yet proved as a group, but reasonably anticipated), this resultreads(3)(1 4 6)(2 5) • (1 2 5 4)(3 6) = (1)(5)(2 4 3 6).We note, for practice, that φ has order 6 (the LCM of 1, 3, and 2) and that theproduct φ • θ has order 4.The number of elements in the group S n of all permutations of degree n isn! and clearly increases very rapidly as n increases. Fortunately, to illustrate theessential features of permutation groups it is sufficient to consider the case n =3,which involves only six elements. They are as follows (with labelling which thereader will by now recognise as anticipatory):I = (1)(2)(3) A =(123) B =(132)C = (1)(2 3) D = (3)(1 2) E = (2)(1 3)It will be noted that A and B have order 3, whilst C, D and E have order 2. Asperhaps anticipated, their combination products are exactly those correspondingto table 28.8, I, C, D and E being their own inverses. For example, putting in allsteps explicitly,D • C{a bc} = (3)(1 2) • (1)(2 3){a bc}= (3)(12){a cb}= {c ab}=(321){a bc}=(132){a bc}= B{a bc}.In brief, the six permutations belonging to S 3 form yet another non-Abelian groupisomorphic to the rotation–reflection symmetry group of an equilateral triangle.1058