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Symmetry Groups

Source lecture(s): PHY622 Lec7

Intuition

Symmetry operations form groups—sets closed under composition with inverses and an identity.

Formal Definition

A group \(G\) is a set with a binary operation satisfying closure, associativity, identity, and inverses.

Mathematical Formulation

\(C_n\): cyclic group of order \(n\). \(D_n\): dihedral group of order \(2n\). \(S_n\): permutation group.

Derivation

Verify closure (product of symmetries is a symmetry), associativity, identity (do nothing), and inverses (undo the operation).

Worked Example

The rotations of an equilateral triangle form \(C_3\); including reflections gives \(D_3\).

Common Mistakes

  • Counting a non-invertible operation as group element.
  • Assuming all groups are Abelian.

Quiz

Q1. Is every subgroup of an Abelian group Abelian?

Answer

Yes.