A Brief Look at Group Theory

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

The combination of any two members of a group must yield another member of the group (closure). 

The group contains the identity, E, multiplication by which commutes with all other members of the group (EA = AE) (identity). 

Every member of the group has a reciprocal such that B B 1 = B 1 B = E where the reciprocal is also a member of the group (inverse). 

it is apparent that reflection through the xz plane, indicated by transforms H into H . More precisely, we could say that H and H are interchanged by reflection. Because the z-axis contains a C2 rotation axis, rotation about the z-axis of the molecule by 180° will take H into H and H into H, but with the halves of each interchanged with respect to the yz plane. The same result would follow from reflection through the xz plane followed by reflection through the yz plane. Therefore, we can represent this series of symmetry operations in the following way  

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