Generators of the symmetric group

Question

I know that for $n\geq2$, $S_n$ (the symmetric group of $n$ symbols) can be generated by only two elements, among which one is a $n$-cycle and other one is a transposition.

But is it true that for any non identity $x\in S_n$ there exist an element $y \in S_n$, such that $S_n$ can be generated by $x$ and $y$?

Please help..

Answer 1

If we avoid following two things, then answer is "interestingly" yes!

• $x\neq 1$
• $n\neq 4$.

Thus, we "state" the following theorem (Ref.- Problems in Group Theory: Dickson):

Theorem [S. Picard]: If $n\neq 4$, then given any $x\in S_n$ with $x\neq 1$, there exists $y\in S_n$ such that $x$ and $y$ generate $S_n$.

Sophie Piccard, Sur les Bases du Groupe Symetrique et du Groupe Alternant, Math. Ann. 116 (1939), pp. 752-767.

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However, I have not read the original proof, and I didn't find alternate reference for this. I can only give "reference" for it.