Is there any normal subgroup of $S_n$?

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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.

hint: the most famous non-trivial subgroup of $S_n$ is normal. — , Jan 25 '13 at 02:16
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 02:19
I thought of posting an answer, but Henry Horton's comment covers it. — Michael Hardy, Jan 25 '13 at 02:30
@Maths Lover: You should have asked for atleast 'non-trivial' subgroup — Aang, Jan 25 '13 at 02:31
@Avatar , how ? what i know that proper subgroup means nontrivial subgroup , can you explain more ? — FNH, Jan 25 '13 at 02:39
@Avatar: usually, it means not entire group nor trivial subgroup (at least across 3 books I've read). Could have been defined differently in your source, obviously. — gnometorule, Jan 25 '13 at 02:40
A common convention is that a proper subgroup of $G$ is a subgroup $H \neq G$, while a non-trivial subgroup of $G$ is a subgroup $H \neq { e }$. But I am sure there are different conventions floating about, and most mathematicians are a bit cavalier about these issues since it's usually (?) clear from context what is meant. — Michael Joyce, Jan 25 '13 at 02:44

amWhy · Accepted Answer · 2013-01-26T01:30:32.977

5

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 $\,S_n$.

edited Jan 26 '13 at 01:30

answered Jan 25 '13 at 02:36

amWhy

thanks . your answer is excellent :) – FNH 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 . – FNH Jan 25 '13 at 23:33

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