Question Number 193871 by Mastermind last updated on 21/Jun/23

Commented by talminator2856792 last updated on 23/Jun/23

Answered by Rajpurohith last updated on 22/Jun/23
![(8) Clearly θ=(18)(17)(16)(15)(14)(13)(12) which shows θ is an odd permutation. hence sgn(θ)=−1. ■ (9) S_n denotes the set of all permutations of a set of n elements say A={1,2,3,...n}. [you know that S_n forms a group under function composition.] let σ∈S_n , so σ is a bijection. How many choices do you have for σ(1)? You have n choices. And for σ(2)? It must be (n−1) choices. because you cannot choose the one you choosed for σ(1). [Suppose you choose σ(2)=σ(1) , ⇒2=1 as σ is bijective.] so you cannot. similarly there are (n−2) choices for σ(3). . . . There are 2 choices for σ(n−1) And only one choice for σ(n). By fundamental principle of counting, we have n(n−1)(n−2)...2.1=n! such permutations on n letters and they are the only permutations. ⇒∣S_n ∣=n! ■ (10)Let β∈S_n . Suppose β is even . ⇒βoβ^(−1) =Id ⇒sgn(βoβ^(−1) )=sgn(Id)=1 Suppose β^(−1 ) is odd ⇒βoβ^(−1) is odd ⇒sgn(βoβ^(−1) )=−1, a contradiction. Hence β^(−1 ) is not odd ⇒β^(−1) must be even. ⇒sgn(β^(−1) )=1=sgn(β) Similar argument for odd. ■](https://www.tinkutara.com/question/Q193888.png)
Commented by Mastermind last updated on 22/Jun/23

Commented by York12 last updated on 22/Jun/23

Commented by Rajpurohith last updated on 22/Jun/23

Commented by talminator2856792 last updated on 23/Jun/23

Commented by York12 last updated on 23/Jun/23

Commented by Rajpurohith last updated on 24/Jun/23

Commented by York12 last updated on 23/Jul/23
