Let \(\sigma\) and \(\tau\) be the permutations of degree 4 defined by \(\sigma 1 = 2\), \(\sigma 2 = 3\), \(\sigma 3 = 4\), \(\sigma 4 = 1\), \(\tau 1 = 3\), \(\tau 2 = 1\), \(\tau 3 = 2\), \(\tau 4 = 4\).
Is \(\sigma\) odd or even? Is \(\tau\) odd or even?
Find \(\sigma\tau\) and \(\tau\sigma\).
Solution
In cycle notation, \[ \sigma=(1234),\qquad \tau=(132). \] A cycle of length \(r\) is a product of \(r-1\) transpositions. Hence \(\sigma\) is odd and \(\tau\) is even.
Using the convention \((\sigma\tau)(i)=\sigma(\tau(i))\), direct evaluation gives \[ \sigma\tau:\quad 1\mapsto4,\ 2\mapsto2,\ 3\mapsto3,\ 4\mapsto1, \] so \[ \sigma\tau=(14). \] Similarly, \[ \tau\sigma:\quad 1\mapsto1,\ 2\mapsto2,\ 3\mapsto4,\ 4\mapsto3, \] so \[ \tau\sigma=(34). \] In particular \(\sigma\tau\ne\tau\sigma\).