Mathematical Methods for Physics and Engineering - Matematica.NET

GROUP THEORYI R R ′ K L MI I R R ′ K L MR R R ′ I M K LR ′ R ′ I R L M KK K L M I R R ′L L M K R ′ I RM M K L R R ′ ITable 28.7 The group table for the two-dimensional symmetry operations onan equilateral triangle.the more self-evident results given in (28.12). A number of things may be noticedabout this table.(i) It is not symmetric about the leading diagonal, indicating that some pairsof elements in the group do not commute.(ii) There is some symmetry within the 3×3 blocks that form the four quartersof the table. This occurs because we have elected to put similar operationsclose to each other when choosing the order of table headings – the tworotations (or three if I is viewed as a rotation by 0π/3) are next to eachother, and the three reflections also occupy adjacent columns and rows.We will return to this later.That two groups of the same order may be isomorphic carries over to non-Abelian groups. The next two examples are each concerned with sets of sixobjects; they will be shown to form groups that, although very different in naturefrom the rotation–reflection group just considered, are isomorphic to it.We consider first the set M of six orthogonal 2 × 2 matrices given by)) ( )−1I =( 1 00 1( ) −1 0C =0 1A =D =(−12√32− √ 32− 1 2( 12− √ 32− √ 32− 1 2)B =E =− √ 32√232− 1 2( √1 32 2√32− 1 2) (28.13)the combination law being that of ordinary matrix multiplication. Here we useitalic, rather than the sans serif used for matrices elsewhere, to emphasise thatthe matrices are group elements.Although it is tedious to do so, it can be checked that the product of anytwo of these matrices, in either order, is also in the set. However, the result isgenerally different in the two cases, as matrix multiplication is non-commutative.The matrix I clearly acts as the identity element of the set, and during the checkingfor closure it is found that the inverse of each matrix is contained in the set, I,C, D and E being their own inverses. The group table is shown in table 28.8.1054