Exercise · ZW-SYMMETRIC-GROUP-007 · ★★☆ Intermediate
Which statement about \(S_n\) is true?
Choose one option.
Show answer
- C. \(S_n\) has \(n!\) elements for every \(n\ge1\)
Explanation
A permutation is determined by choosing a distinct image for each element, giving \(n!\) possibilities; the other statements are false (the order of composition matters from \(n=3\) on and the identity is always neutral).
Review the concept: Symmetric group
The symmetric group \(S_n\) is the set of bijections \(\sigma\) of \(\{1,\dots,n\}\) onto itself, with composition \(\sigma\tau\) as the operation.
\[(\sigma\tau)(i)=\sigma(\tau(i))\]
More practice problems: symmetric group
- ZW-SYMMETRIC-GROUP-001 How many elements does S_4 have?
- ZW-SYMMETRIC-GROUP-002 Write a permutation of S_3 as the list [σ(1),σ(2),σ(3)]. If σ=[2,3,1] and τ=[2,1,3], what is στ (τ…
- ZW-SYMMETRIC-GROUP-003 What is the identity element of S_5?
- ZW-SYMMETRIC-GROUP-004 If σ=[2,3,1,4] in S_4 (list [σ(1),…,σ(4)]), what is σ^-1?
- ZW-SYMMETRIC-GROUP-005 What is the smallest n for which S_n is not abelian?
- ZW-SYMMETRIC-GROUP-006 How many permutations σ∈ S_5 satisfy σ(1)=1?
- ZW-SYMMETRIC-GROUP-008 For which n does S_n have exactly 120 elements?
- ZW-SYMMETRIC-GROUP-009 In S_4 let σ=[2,1,3,4] and τ=[1,3,2,4]. What is (στ)(2)?