GROUP THEORY
The multiplication table for this set of six functions has all the necessary proper-
ties to show that they form a group. Further, if the symbolsf 1 ,f 2 ,f 3 ,f 4 ,f 5 ,f 6 are
replaced byI, A, B, C, D, Erespectively the table becomes identical to table 28.8.
This justifies our earlier claim that this group of functions, with argument sub-
stitution as the law of combination, is isomorphic to the group of reflections and
rotations of an equilateral triangle.
28.4 Permutation groupsThe operation of rearrangingndistinct objects amongst themselves is called a
permutationof degreen, and since many symmetry operations on physical systems
can be viewed in that light, the properties of permutations are of interest. For
example, the symmetry operations on an equilateral triangle, to which we have
already given much attention, can be considered as the six possible rearrangements
of the marked corners of the triangle amongst three fixed points in space, much
as in the diagrams used to compute table 28.7. In the same way, the symmetry
operations on a cube can be viewed as a rearrangement of its corners amongst
eight points in space, albeit with many constraints, or, with fewer complications,
as a rearrangement of its body diagonals in space. The details will be left until
we review the possible finite groups more systematically.
The notations and conventions used in the literature to describe permutationsare very varied and can easily lead to confusion. We will try to avoid this by using
lettersa,b,c,...(rather than numbers) for the objects that are rearranged by a
permutation and by adopting, before long, a ‘cycle notation’ for the permutations
themselves. It is worth emphasising that it is thepermutations,i.e.theactsof
rearranging, and not the objects themselves (represented by letters) that form
the elements of permutation groups. The complete group of all permutations of
degreenis usually denoted bySnor Σn. The number of possible permutations of
degreenisn!, and so this is the order ofSn.
Suppose the ordered set of six distinct objects{abcdef}is rearranged bysome process into{befadc}; then we can represent this mathematically as
θ{abcdef}={befadc},whereθis a permutation of degree 6. The permutationθcan be denoted by
[256143],sincethefirstobject,a, is replaced by the second,b, the second
object,b, is replaced by the fifth,e, the third by the sixth,f,etc.Theequation
can then be written more explicitly as
θ{abcdef}=[256143]{abcdef}={befadc}.Ifφis a second permutation, also of degree 6, then the obvious interpretation of
the productφ•θof the two permutations is
φ•θ{abcdef}=φ(θ{abcdef}).