Question

Prove that$\left(12\right)$ and $\left(123\cdots n\right)$generate$S_n$ .

Short answer

It is proved that $\left(12\right)$and $\left(123\cdots n\right)$generates $S_n$.

Step-by-step solution

Exercise 36

From exercise 36, we have$\sigma \tau {\sigma }^{-1}=\left(\sigma \left(a_1\right)\sigma \left(a_2\right)\dots \sigma \left(a_k\right)\right)$ . This implies that for all $0⩽i⩽n-2$that $\left(1\dots n\right)\left(12\right){\left(1\dots n\right)}^{-1}=\left(1+i,2+i\right)$.Therefore, contains all transpositions $\left(i,i+1\right)$for$0⩽i<n$ .

Prove the result

For all $1<k⩽n$, we have $\left(1\dots k\right)=\left(12\right)\left(23\right)\dots \left(k-1k\right)$, this implies that . From exercise 36, we have for all $0⩽i<n$,

We know that from exercise 41, the transpositions $\left(12\right),\left(13\right),\left(14\right),\dots ,\left(1n\right)$generates $S_n$. Therefore, .

Hence, it is proved that $\left(12\right)$and $\left(123\cdots n\right)$generate $S_n$.