Introduction to Groups Symmetries of an Equilateral Triangle Rotational Symmetry β’ Let π π be a counterclockwise rotation of π degrees. β’ What are the possible symmetry rotations of an equilateral triangle? π 0 , π 120 , and π 240 Rotations π 0 π 120 π 240 Reflective Symmetry What are the possible reflections of an equilateral triangle? π π πΏ Composition We can compose (or combine or multiply) the 6 symmetries, but this will always be equivalent to one of the original 6. π π 120 Which of the original 6 is this equivalent to? πΏ This is the same as πΏ. So we can say π 120 β π = πΏ Summary of Symmetry π 0 π π 120 π π 240 πΏ Cayley Table We can summarize these computations in a Cayley table (or operation table.) π π πβπ π 0 π 120 π 240 π π πΏ π 0 π 0 π 120 π 240 π π πΏ π 120 π 120 π 240 π πΏ π π 240 π 0 πΏ π π π π 240 π π 0 π 120 πΏ π π 0 π 240 π 120 π π π πΏ π 120 π 0 π 240 πΏ πΏ π π π 240 π 120 π 0 πβπ π 0 π 120 π 240 π π πΏ π 0 π 0 π 120 π 240 π π πΏ π 120 π 120 π 240 π πΏ π π 240 π 0 πΏ π π π π 240 π π 0 π 120 πΏ π π 0 π 240 π 120 π π π πΏ π 120 π 0 π 240 πΏ πΏ π π π 240 π 120 π 0 This is an example of a group. It is called the βDihedral Group of order 6β and is denoted by π·3 . Notice: 1. Each composition is one of the original 6 symmetries. (Closure: If A and B are in π·3 , then so is AB.) 2. π 0 acts as an identity: π 0 β π΄ = π΄ β π 0 = π΄ 3. Every element has an inverse: π β π = π 0 and π 120 β π 240 = π 240 β π 120 = π 0 4. The compositions in π·3 are associative: 5. Is the operation commutative? π΄π΅ πΆ = π΄(π΅πΆ) No, this is an example of a nonabelian group. If a group has commutativity, it is an abelian group.
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