# 3.3: Successive Operations

- Page ID
- 32160

Sometimes, new symmetry operations form by carrying out two or more simpler operations successively to result in an indistinguishable configuration. For example, the improper rotation axis that was introduced in the last section results from rotation about an axis of rotation (C_{n}) followed by reflection about a plane perpendicular to that axis (\(\sigma\)_{h}).

\[S_{n}=C_{n}\times\sigma_{h}\]

Note also that C_{n} in this case need not be a symmetry element for the molecule. For example, staggered ethane has an S_{6} symmetry element although it does not have a C_{6}. Another example of a symmetry element that results from a combination of two different elements is the inversion center (i), which results from a C_{2} rotation followed by a reflection about a plane perpendicular to that axis (\(\sigma\)_{h}).

\[i=C_{2}\times\sigma_{h}\]

Note that an inversion center is a special case of an improper rotation axis because it results when, in the first equation, \(n=2\). That is,

\[i=S_{2}\]

In fact, any symmetry operation can be carried out multiple time in a row. For example, when BH_{3} is rotated twice by 120°, the two-step operation can be symboled by \(C_{3}^{2}\). When the C_{3} operation is performed a third time, the molecule returns to its original configuration; i.e,

\[C_{3}^{3}=E\]

In general,

\[C_{n}^{n}=E\]