Worksheet: Symmetry Operations

In this worksheet, we will practice defining a symmetry operation and describing the effect of inversion, reflection, rotation, and rotoinversion symmetries.


From which of the following situations does an improper rotation axis, 𝑆, arise?

  • ATwo proper rotation axes
  • BTwo mirror planes
  • CA proper rotation axis and a mirror plane
  • DA mirror plane and an inversion
  • EA mirror plane and the identity operation


Which of the following properties is not a prerequisite for symmetry elements within the same symmetry group?

  • AAssociativity (𝐴(𝐵𝐶)=(𝐴𝐵)𝐶)
  • BIdentity (The symmetry element times its identity, 𝐸, remains unchanged.)
  • CContinuity (If a molecule contains an 𝑛 axis, then it also contains axes of 𝑛1, , 𝑛2.)
  • DInverses (There must be an inverse symmetry element that returns the molecule to its original state; 𝐴×𝐴=𝐸.)
  • EClosure (If 𝐴 and 𝐵 belong to the group, and 𝐴×𝐵=𝐶, then 𝐶 also belongs to the group.)

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