Generators for $S_n$

finite-groups

This problem is from Dixon's book telling that; if $x$ is any nontrivial element of the symmetric group $S_n$ and $n≠4$, then there exists an element $y\in S_n $ such that $S_n=\langle x,y\rangle$. Honestly,the reference which Dixon referred to, is out of my hand. I can see this fact through $$S_3:=\{(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) \}.$$ Please make sparks for me . Thanks.


Turns out to be problem 2.63 (which is starred, indicating its difficulty) in Problems in Group Theory by John D. Dixon, Dover Publications.

The solution provided is a simple reference. Unfortunately, the name of the author is misspelled in the reference:

Sophie Piccard, Sur les bases du groupe symétrique et du groupe alternant, Math. Ann. 116 (1939), pp. 752-767.

(The book lists the name incorrectly as "S. Picard")

The paper seems to be available through the Göttinger Digitalisierungszentrums, by going here and moving the appropriate page. I haven't had time to look through it.

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