Permutation groups
Objective. Compute cycles, parity and composition.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Permutation groups study bijections of a finite set under composition, with the symmetric group Sₙ containing all n! permutations of {1,…,n}. Each permutation decomposes into disjoint cycles, and its parity (even or odd) is determined by writing it as a product of transpositions: an even number of transpositions gives an even permutation. Composition of cycles is computed right to left, applying the rightmost permutation first. For σ = (1 2 3) and τ = (2 3) in S₃, στ sends 1→1→2... computing carefully: apply τ then σ, so 1→1→2, 2→3→1, 3→2→3, giving στ = (1 2).
Worked example
Compute the product of the permutations (1 2)(3 4) and (1 3) in cycle notation, applying the rightmost first.
- 1Apply (1 3) first: 1→3, 2→2, 3→1, 4→4.
- 2Then apply (1 2)(3 4) to each result: 3→4, 2→1, 1→2, 4→3.
- 3Combining, the overall map sends 1→4, 2→1, 3→2, 4→3.
- 4Writing this as a cycle starting from 1 gives (1 4 3 2).
Answer: (1 4 3 2)
Concept mastery
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Practice
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Start practisingCommon mistake. Students compose permutations left to right instead of right to left (or vice versa depending on convention), reversing the intended order and producing the wrong resulting permutation.
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