Inverses of operations

Every symmetry operation in the group has an inverse operation that is also a member of the group. In this context, the word inverse should not be confused with inversion. The mathematical inverse of an operation is its reciprocal, such that A A = A A = , where the symbol A represents the inverse of operation A. The identity element will always be its own inverse. Likewise, the inverse of any reflection operation will always be the original reflection. The inversion operation (/) is also its own inverse. The inverse of a C proper rotation (counterclockwise) will always be the symmetry operation that is equivalent to a C rotation in the opposite direction (clockwise). No two operations in the group can have the same inverse. The list of inverses for the symmetry operations in the ammonia symmetry group are as follows ... 

This expression no longer contains the inverse of operators but the inverse of matrix representations of the operators. Prom comparison of Eq. (3.149) and Eq. (3.157) we may conclude that a matrix representation of the superoperator resolvent is given by... 

Note that Eq. (1.258) defines the total electric field in term of itself, and thus it can be solved by iterative methods or by defining a direct inversion of operators ... 

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