Question

Prove that the transpositions$\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$generate$S_n$.

Short answer

It is proved that the transpositions $\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$generate $S_n$.

Step-by-step solution

Given

Let us assume that $\sigma \in S_n$be a k-cycle and let $\sigma =\left(a_1{a}_2\dots a_k\right)$be a representation where is the minimal element in the set $\left\{a_1,\dots ,a_k\right\}$$\left\{a_1,\dots ,a_k\right\}$.

Prove the result

Assume that $a_1=1$ then $\sigma =\left(1a_k\right)\dots \left(1a_3\right)\left(1a_2\right)$ and if $a_1\ne 1$ then .

$\sigma =\left(1a_1\right)\left(1a_k\right)\dots \left(1a_2\right)\left(1a_1\right)$

As every element of is a product of cycles, this implies that for all -cycles are in the set generated by the transpositions $\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$. We can conclude that $\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$ generate all of $S_n$.

Hence,it is proved that the transpositions$\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$ generate $S_n$.