Disjoint cycles commute in symmetric groups


by Patrick Stevens Jun 14 2016 updated Jun 14 2016

In cycle notation, if two cycles are disjoint, then they commute.

[summary: In a symmetric group, if we are applying a collection of permutations which are each disjoint cycles, we get the same result no matter the order in which we perform the cycles.]

Consider two cycles $~$(a_1 a_2 \dots a_k)$~$ and $~$(b_1 b_2 \dots b_m)$~$ in the Symmetric group $~$S_n$~$, where all the $~$a_i, b_j$~$ are distinct.

Then it is the case that the following two elements of $~$S_n$~$ are equal:

Indeed, $~$\sigma(a_i) = (b_1 b_2 \dots b_m)[(a_1 a_2 \dots a_k)(a_i)] = (b_1 b_2 \dots b_m)(a_{i+1}) = a_{i+1}$~$ (taking $~$a_{k+1}$~$ to be $~$a_1$~$), while $~$\tau(a_i) = (a_1 a_2 \dots a_k)[(b_1 b_2 \dots b_m)(a_i)] = (a_1 a_2 \dots a_k)(a_i) = a_{i+1}$~$, so they agree on elements of $~$(a_1 a_2 \dots a_k)$~$. Similarly they agree on elements of $~$(b_1 b_2 \dots b_m)$~$; and they both do not move anything which is not an $~$a_i$~$ or a $~$b_j$~$. Hence they are the same permutation: they act in the same way on all elements of $~$\{1,2,\dots, n\}$~$.

This reasoning generalises to more than two disjoint cycles, to show that disjoint cycles commute.