- Show that disjoint cycles commute.
Solution
Let \[ \sigma=(a_1\,\dots\,a_r),\qquad \tau=(b_1\,\dots\,b_s) \] be disjoint cycles.
1. For every point \(x\), one has\[ \sigma\tau(x)=\tau\sigma(x). \]
There are three cases.
If \(x\) lies in the support of \(\sigma\), then \(x\) is fixed by \(\tau\) because the supports are disjoint. Also \(\sigma(x)\) still lies in the support of \(\sigma\), so it is fixed by \(\tau\). Hence \[ \sigma\tau(x)=\sigma(x)=\tau\sigma(x). \]
If \(x\) lies in the support of \(\tau\), the same argument with the roles reversed gives \[ \sigma\tau(x)=\tau(x)=\tau\sigma(x). \]
If \(x\) lies in neither support, both cycles fix \(x\), so both composites fix \(x\).
2. Therefore \(\sigma\tau=\tau\sigma\).
Two permutations are equal when they agree on every point, and 1 shows exactly that.