By F. Kirk

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N t1 t2 . tn s 1 s 2 . sn 1 2 . n = . t 1 t 2 . tn The permutations are found to have the following properties: 1. } is itself a permutation of the set. (Closure) 2. The product obeys associativity: (πkπj)πi = πk(πjπi), (not generally commutative). 3. An identity permutation exists. 4. An inverse permutation exists: s 1 s 2 . . sn π = -1 1 2 . . n such that ππ -1 = π-1π = identity permutation. 7 Cayley’s theorem: Every finite group is isomorphic to a certain permutation group. Let Gn ={g 1, g 2, g 3, .

A group with no proper (non-trivail) invariant subgroups is said to be simple (atomic). If none of the proper invariant subgroups of a group is abelian, the group is said to be semisimple. An invariant subgroup Hm and its cosets form a group under multiplication called the factor group (written Gn /Hm ) of Hm in Gn . These formal aspects of Group Theory can be illustrated by considering the following example: 46 The group D3 = {e, a, a 2, b, c, d} ~ S 3 = {123, 312, 231, 132, 321, 213}. C3 is a subgroup of S3 : C3 = H 3 = {e, a, a 2} = {123, 312, 231}.

In component form, we have L z cl = xp y − yp x, where px and p y are the cartesian components of p. The transition between Classical and Quantum Mechanics is made by replacing p x by −i(h/2π)∂/∂x (a differential operator) p y by −i(h/2π)∂/∂y (a differential operator),where h and is Planck’s constant. We can therefore write the quantum operator as Lz Q = −i(h/2π)(x∂/∂y − y∂/∂x) = −i(h/2π)X and therefore X = iLz Q/(h/2π), and δxi = Xxi δφ = (2πiLz Q/h)xi δφ, i = 1,2. Let an arbitrary, continuous, differentiable function f(x, y) be transformed under the infinitesimal changes x´ = x − yδφ y´ = y + xδφ .