6
$\begingroup$

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..

$\endgroup$
7
$\begingroup$

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.


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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.