Exercise · ZW-CYCLE-OF-A-PERMUTATION-001 · ★☆☆ Basic
In \(S_5\), the cycle \((1\ 3\ 5)\) sends \(3\) to…
Choose one option.
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- D. \(5\)
Explanation
A cycle sends each entry to the next one: \(1\mapsto3\mapsto5\mapsto1\). So \(3\) goes to \(5\); \(4\) does not appear in the cycle and would stay fixed.
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-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:
- ZW-CYCLE-OF-A-PERMUTATION-009 In S_4, the square of the cycle (1\ 2\ 3\ 4) (the cycle applied twice) is…