Prove that every permutation in $S_n$ may be written in terms of $(1,2)$ and one non-trivial element.

Question

Need which property of algebra, or otherwise; to prove it.

Answer 1

Hint $S_n=\langle (12),(12\dots n)\rangle $

Proof:

It suffices to show you can get any transposition ($2$-cycle).

Note $(12\dots n)^{k}(12)(12\dots n)^{-k}=(k+1 k+2)$.

From there you can get the rest: $(12)(23)(12)=(13)$ etc.

Answer 2

Let $s_i=(i,i+1)$ where $1\leq i$ and we think this definition$\mod n$, zero being $n$. Let $\alpha=(1,2,3,...,n)$. Then it can be checked that $$\alpha s_i\alpha^{-1}=s_{i+1}.$$ Hence, all $s_i$s are generated by $s_1$ and $\alpha$.

But on the other hand, we have the identities $s_j(i,j)s_j=(i,j+1)$ which implies that all 2-cycles are generated by $s_i$s. Hence, all 2-cycles are generated by $s_1$ and $\alpha$.

But, we know that the symmetric group is generated by 2-cycles.

Hence, the group $S_n$ is generated by $s_1$ and $\alpha$.