Exercise · ZW-CYCLE-OF-A-PERMUTATION-009 · ★★★ Advanced
In \(S_4\), the square of the cycle \((1\ 2\ 3\ 4)\) (the cycle applied twice) is…
Choose one option.
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- A. \((1\ 3)(2\ 4)\)
Explanation
Applied twice, \(1\) goes to \(3\) and \(3\) to \(1\); likewise \(2\) goes to \(4\) and \(4\) to \(2\). The square of a 4-cycle is not a cycle but two disjoint cycles of length 2.
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-008 In S_6, the permutation with list [2,1,4,5,3,6] is written as a product of disjoint cycles as:
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