Exercise · ZW-CYCLE-OF-A-PERMUTATION-008 · ★★★ Advanced
In \(S_6\), the permutation with list \([2,1,4,5,3,6]\) is written as a product of disjoint cycles as:
Choose one option.
Show answer
- B. \((1\ 2)(3\ 4\ 5)\)
Explanation
Follow each element: \(1\mapsto2\mapsto1\) and \(3\mapsto4\mapsto5\mapsto3\), while \(6\) stays fixed and is not written. The option with \((3\ 5\ 4)\) runs the cycle backwards.
Review the concept: Cycle of a permutation
A cycle of length \(k\) in \(S_n\) is a permutation sending \(a_1\mapsto a_2\mapsto\cdots\mapsto a_k\mapsto a_1\) and fixing all other elements.
\[(a_1\ a_2\ \cdots\ a_k)\]
More practice problems: cycle of a permutation
- ZW-CYCLE-OF-A-PERMUTATION-001 In S_5, the cycle (1\ 3\ 5) sends 3 to…
- ZW-CYCLE-OF-A-PERMUTATION-002 What is the length of the cycle (2\ 5\ 3\ 1)?
- ZW-CYCLE-OF-A-PERMUTATION-003 Which of these cycles is the same as (1\ 2\ 3\ 4)?
- ZW-CYCLE-OF-A-PERMUTATION-004 What is the inverse of the cycle (1\ 2\ 3\ 4)?
- ZW-CYCLE-OF-A-PERMUTATION-005 In S_3, what is (1\ 2)(2\ 3) (the right-hand cycle is applied first)?
- ZW-CYCLE-OF-A-PERMUTATION-006 Which statement about cycles in S_n is true?
- ZW-CYCLE-OF-A-PERMUTATION-007 How many distinct cycles of length 3 are there in S_4?
- ZW-CYCLE-OF-A-PERMUTATION-009 In S_4, the square of the cycle (1\ 2\ 3\ 4) (the cycle applied twice) is…
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