Group theory

28.4 PERMUTATION GROUPSSuppose that φ is the permutation [4 5 3 6 2 1]; thenφ • θ{a b c d e f} = [4 5 3 6 2 1][2 5 6 1 4 3]{a b c d e f}= [4 5 3 6 2 1]{b e f a d c}= {a d f c e b}= [1 4 6 3 5 2]{a b c d e f}.Written in terms of the permutation notation this result is[4 5 3 6 2 1][2 5 6 1 4 3] = [1 4 6 3 5 2].A concept that is very useful for working with permutations is that of decompositioninto cycles. The cycle notation is most easily explained by example. Forthe permutation θ given above:the 1st object, a, has been replaced by the 2nd, b;the 2nd object, b, has been replaced by the 5th, e;the 5th object, e, has been replaced by the 4th, d;the 4th object, d, has been replaced by the 1st, a.This brings us back to the beginning of a closed cycle, which is convenientlyrepresented by the notation (1 2 5 4), in which the successive replacementpositions are enclosed, in sequence, in parentheses. Thus (1 2 5 4) means 2nd→ 1st, 5th → 2nd, 4th → 5th, 1st → 4th. It should be noted that the objectinitially in the first listed position replaces that in the final position indicated inthe bracket – here ‘a’ is put into the fourth position by the permutation. Clearlythe cycle (5 4 1 2), or any other that involved the same numbers in the samerelative order, would have exactly the same meaning and effect. The remainingtwo objects, c and f, are interchanged by θ or, more formally, are rearrangedaccording to a cycle of length 2, a transposition, represented by (3 6). Thus thecomplete representation (specification) of θ isθ = (1 2 5 4)(3 6).The positions of objects that are unaltered by a permutation are either placed bythemselves in a pair of parentheses or omitted altogether. The former is recommendedas it helps to indicate how many objects are involved – important whenthe object in the last position is unchanged, or the permutation is the identity,which leaves all objects unaltered in position! Thus the identity permutation ofdegree 6 isI = (1)(2)(3)(4)(5)(6),though in practice it is often shortened to (1).It will be clear that the cycle representation is unique, to within the internalabsolute ordering of the numbers in each bracket as already noted, and that1057