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Citation: Weina Wang, Fengyi Liu, Wenliang Wang. Introduction to a Concise Deduction for the Independence of the Rotation-Reflection Axis S₄ and the Rotation-Inversion Axis I₄[J]. University Chemistry, ;2023, 38(5): 249-253. doi: 10.3866/PKU.DXHX202207007 shu

Introduction to a Concise Deduction for the Independence of the Rotation-Reflection Axis S₄ and the Rotation-Inversion Axis I₄

School of Chemistry and Chemical Engineering, Shaanxi Normal University, Xi'an 710119, China

Corresponding author: Wenliang Wang, wlwang@snnu.edu.cn
Received Date: 1 July 2022

The point groups of molecular and crystal symmetry are among the important contents in structural chemistry, and they constitute a challenge in teaching and learning. Students are often confused regarding why only the rotation-reflection axis S₄ and the rotation-inversion axis I₄ are independent, while other rotation-reflection and rotation-inversion axes are not, or why S₄ is used in describing molecular symmetry, while I₄ is used in describing crystal symmetry. In this article, starting from the generators of the cyclic group, we obtained the symmetrical operation elements of the point groups S_n and I_n, respectively. According to whether S_n (I_n) can be replaced by another symmetric element, or whether the symmetric operating group can be represented as the direct product of two subgroups, we deducted that only S₄and I₄ are independent when n ≤ 6. This deduction avoids the tediousness and abstraction of graphical illustrations. It is helpful for students to understand the independence of symmetrical elements from the definition derivation, and convenient to promote in teaching.
- Symmetry operation group,
- Rotation-reflection axis S₄,
- Rotation-inversion axis I₄,
- Independence
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Introduction to a Concise Deduction for the Independence of the Rotation-Reflection Axis S₄ and the Rotation-Inversion Axis I₄