As the title says: how can I find the subgroups of $S_n$? I know that $S_n = <(1 2), (1 2 .. n)>$ and $S_n = <(1 i): i = 2, 3, .. n>$. But how do I find all subgroups? And how do I know their orders?


marked as duplicate by Namaste abstract-algebra Apr 11 '18 at 12:31

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  • $\begingroup$ E.g.: How to find all the subgoups of $S_3$ $\endgroup$ – Namaste Apr 11 '18 at 12:28
  • $\begingroup$ See also Is there any efficient algorithm for finding subgroups of a given finite group?, which addresses also, $S_n$, a finite group of size $n$. $\endgroup$ – Namaste Apr 11 '18 at 12:32
  • $\begingroup$ Wilson's "Finite Simple Groups" deals with the the subgroups of $S_n$ and much else of interest besides. amazon.co.uk/Finite-Simple-Groups-Graduate-Mathematics/dp/… $\endgroup$ – Mark Bennet Apr 11 '18 at 12:41
  • $\begingroup$ @amWhy I have voted to reopen this, because the specific $S_n$ case has been studied, and is more tractable, as you will see from the reference I posted in my other comment (check the index - there is a chapter on the problem - also coverage in Dixon and Mortimer "Permutation Groups"). I imagine the analysis would be a stretch for OP though - but an answer to the $S_n$ case would, I think, be valuable for the site. $\endgroup$ – Mark Bennet Apr 11 '18 at 12:47
  • $\begingroup$ @amWhy Further to above I see am=nother duplicate now listed, which does cover the case. $\endgroup$ – Mark Bennet Apr 11 '18 at 12:50