# Show than $(1,k+1), (1,2,3,…,n)$ generate the group $S_n$ if and only if $k$ and $n$ are coprime

I am able to prove constructively that if $$(k,n)=1$$ then we can generate $$S_n$$ but am struggling with the converse.

• Show that the generators preserve the partition into congruence classes mod $(k,n)$. – user10354138 Jun 2 at 21:05

## 1 Answer

Let $$n\geq 2$$ and $$1\leq a < b \leq n$$. Then $$S_n$$ is generated by the transposition $$(ab)$$ and the $$n$$-cycle $$(12\dots n)$$ if and only if $$\gcd(b-a,n)=1$$.

For a complete proof see Theorem 2.8 of Keith Conrad's blurb on Generating Sets.