Symmetric group question

  • Thread starter autre
  • Start date
  • #1
autre
117
0

Homework Statement



[itex]sgn(\sigma)=0\iff\sigma[/itex] is not a bijection.

The Attempt at a Solution



[itex](\rightarrow)[/itex]Let [itex]sgn(\sigma)=0[/itex]. Then, [itex]\Pi_{1\leq i<j\leq n}\frac{\sigma(j)-\sigma(i)}{j-i}=0. [/itex]. For some [itex]i[/itex] and [itex]j[/itex], [itex]i\neq j [/itex], [itex]\sigma(i)=\sigma(j) [/itex]. Thus, [itex]\sigma[/itex] is not an injection.

[itex](\leftarrow)[/itex] [itex]\sigma[/itex] not a bijection [itex]\rightarrow[/itex] it is not an injection, same argument as above, [itex]sgn(\sigma)=0[/itex].

I think I went wrong somewhere. Any ideas?
 
Last edited:

Answers and Replies

Suggested for: Symmetric group question

Replies
11
Views
736
  • Last Post
Replies
6
Views
809
  • Last Post
Replies
4
Views
376
  • Last Post
Replies
16
Views
1K
Replies
6
Views
521
Replies
2
Views
634
Replies
24
Views
2K
Replies
17
Views
558
  • Last Post
Replies
6
Views
592
Top