# Is there any normal subgroup of $S_n$? [duplicate]

Possible Duplicate:
Normal subgroups of $S_N$

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

If yes, give an example.

• 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

## 1 Answer

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 $\,S_n$.
• thanks . your answer is excellent :) – Fawzy Hegab 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 . – Fawzy Hegab Jan 25 '13 at 23:33