Exercise · ZW-CYCLE-OF-A-PERMUTATION-006 · ★★☆ Intermediate
Which statement about cycles in \(S_n\) is true?
Choose one option.
Show answer
- D. Two cycles with no common element commute
Explanation
Disjoint cycles act on different elements and do not interfere, so their product does not depend on the order. Two cycles sharing elements, such as \((1\ 2)\) and \((2\ 3)\), may fail to commute.
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-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:
- 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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