# Show that transposition is an odd permutation

This is in symmetric groups $S_n$. I don't know how to approach this problem. I guess I have to use the definition of $$\mathrm{sgn}(\sigma)=\prod_{1 \leq i < j \leq n} \frac{\sigma(j)-\sigma(i)}{j-i}$$

I know I have to use this on the permutation $(i \ j)$ which represents a transposition of the $i$th and $j$th elements of the permutation, but I don't know how to approach this.

• Do an example for, say, $n=5$. – arctic tern Apr 9 '17 at 19:42
• I know it will give me $-1$ but how can I do it for $n = n$? – The Bosco Apr 9 '17 at 19:45
• Count the number of inversions. – Bernard Apr 9 '17 at 19:51
• I know it is only 1 inversion, but I have to use that formula because if I use the definition that an inversion changes the parity I will not be proving anything. – The Bosco Apr 9 '17 at 19:59