# 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.

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

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
If this is homework, please add the "homework" tag. –  Alex Becker Jan 25 '13 at 2:21
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

• 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 $\,S_n$.