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Possible Duplicate:
Normal subgroups of $S_N$

I wonder if there is any normal proper subgroup of $S_n$?

If yes, give an example.

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    $\begingroup$ hint: the most famous non-trivial subgroup of $S_n$ is normal. $\endgroup$ – user29743 Jan 25 '13 at 2:16
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    $\begingroup$ The alternating group $A_n$ has index $2$ in $S_n$, and index $2$ subgroups are always normal... $\endgroup$ – Henry T. Horton Jan 25 '13 at 2:19
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    $\begingroup$ Sure $H=\{e\}$. $\endgroup$ – JSchlather Jan 25 '13 at 2:24
  • $\begingroup$ I thought of posting an answer, but Henry Horton's comment covers it. $\endgroup$ – Michael Hardy Jan 25 '13 at 2:30
  • $\begingroup$ @Maths Lover: You should have asked for atleast 'non-trivial' subgroup $\endgroup$ – Aang Jan 25 '13 at 2:31
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Certainly, yes.

  • The alternating group $\,A_n \leq S_n\,$ is a normal subgroup of $\,S_n\,$, since its index $\,[S_n : A_n] = 2$,
  • and all subgroups of index $\,2\,$ are normal. (The last link is to a proof of this fact.)
  • Indeed, for all $S_n \;\text{with }\,n\geq 5,\,$ $A_n\,$ is the ONLY normal (non-trivial) subgroup of $\,S_n$.
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  • $\begingroup$ thanks . your answer is excellent :) $\endgroup$ – Fawzy Hegab Jan 25 '13 at 23:31
  • $\begingroup$ You're welcome! $\endgroup$ – amWhy Jan 25 '13 at 23:31
  • $\begingroup$ :D i asked a question about group actions and permuations groups form serval minutes , plz check it and help if you can :) , thanx again . $\endgroup$ – Fawzy Hegab Jan 25 '13 at 23:33

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