Decomposing (1 3 5 2)(2 4 3) into Disjoint Cycles

Abstract Algebra · Axiom Academy

EXAMPLE Decomposing Permutations into Disjoint Cycles Learn to compose cycles, trace elements, and find the disjoint cycle decomposition Excellent work! You've successfully decomposed a permutation into disjoint cycles. Here's what we learned: Composition order matters: Read permutations right to left – apply the rightmost cycle first Trace systematically: For each element, follow it through each cycle in the composition Build the mapping: Record where each element ends up after the full composition Identify cycles: Start with any element and follow its chain until you return to the start Disjoint cycles: Cycles are disjoint when they have no elements in common Calculate order: The order of a permutation equals lcm of all its cycle lengths This technique works for any permutation composition! Practice with different cycles to build confidence. Remember: disjoint cycle decomposition is unique (up to ordering), making it a powerful tool for analyzing permutation structure.

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