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.
Generators for $S_n$
8
$\begingroup$
finite-groups
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0There are some classical sets of generators for the symmetric group. It might be handy to document these somewhere. All transpositions works. A transposition and a suitable n-cycle (which n-cycles work - I know some are easy to prove). A whole conjugacy class of odd permutations will also work (I can get any 3-cycle, by composing a suitable pair, hence $A_n$, and have an odd permutation too). – 2012-05-11
1 Answers
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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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0@Steve: The theorem of Picard in interesting, and thanks for the modern reference. Nice reference. – 2013-05-31