Symmetric group question

  • Thread starter autre
  • Start date
  • #1
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

Related Threads on Symmetric group question

  • Last Post
2
Replies
28
Views
3K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
10
Views
9K
Replies
1
Views
1K
  • Last Post
Replies
3
Views
3K
Replies
1
Views
2K
Replies
4
Views
6K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
5
Views
2K
Top