Mathematical Methods for Physics and Engineering - Matematica.NET

GROUP THEORY1 i −1 −i1 1 i −1 −ii i −1 −i 1−1 −1 −i 1 i−i −i 1 i −11 2 4 31 1 2 4 32 2 4 3 14 4 3 1 23 3 1 2 4Table 28.5 A comparison between tables 28.4 and 28.2(b), the latter with itscolumns reordered.I A B CI I A B CA A B C IB B C I AC C I A BTable 28.6 The common structure exemplified by tables 28.4 and 28.2(b), thelatter with its columns reordered.amount of relabelling (or, equivalently, no allocation of the symbols A, B, C, amongsti, −1, −i) can bring table 28.4 into the form of table 28.3. We conclude that the group{1,i,−1, −i} is not isomorphic to S or S ′ . An alternative way of stating the observation isto say that the group contains only one element of order 2 whilst a group correspondingto table 28.3 contains three such elements.However, if the rows and columns of table 28.2(b) – in which the identity does appeartwice on the diagonal and which therefore has the potential to be equivalent to table 28.4 –are rearranged by making the heading order 1, 2, 4, 3 then the two tables can be comparedin the forms shown in table 28.5. They can thus be seen to have the same structure, namelythat shown in table 28.6.We therefore conclude that the group of four elements {1,i,−1, −i} under ordinary multiplicationof complex numbers is isomorphic to the group {1, 2, 3, 4} under multiplication(mod 5). ◭What we have done does not prove it, but the two tables 28.3 and 28.6 are infact the only possible tables for a group of order 4, i.e. a group containing exactlyfour elements.28.3 Non-Abelian groupsSo far, all the groups for which we have constructed multiplication tables havebeen based on some form of arithmetic multiplication, a commutative operation,with the result that the groups have been Abelian and the tables symmetricabout the leading diagonal. We now turn to examples of groups in which somenon-commutation occurs. It should be noted, in passing, that non-commutationcannot occur throughout a group, as the identity always commutes with anyelement in its group.1052