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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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marked as duplicate by Austin Mohr, Brandon Carter, Micah, Jonas Meyer, Mariano Suárez-Alvarez Jan 25 '13 at 3:18

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

hint: the most famous non-trivial subgroup of $S_n$ is normal. – user29743 Jan 25 '13 at 2:16
The alternating group $A_n$ has index $2$ in $S_n$, and index $2$ subgroups are always normal... – Henry T. Horton Jan 25 '13 at 2:19
Sure $H=\{e\}$. – JSchlather Jan 25 '13 at 2:24
I thought of posting an answer, but Henry Horton's comment covers it. – Michael Hardy Jan 25 '13 at 2:30
@Maths Lover: You should have asked for atleast 'non-trivial' subgroup – Aang Jan 25 '13 at 2:31
up vote 4 down vote accepted

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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thanks . your answer is excellent :) – Maths Lover Jan 25 '13 at 23:31
You're welcome! – amWhy Jan 25 '13 at 23:31
:D i asked a question about group actions and permuations groups form serval minutes , plz check it and help if you can :) , thanx again . – Maths Lover Jan 25 '13 at 23:33

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