# Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$

There is a famous Theorem telling that:

For $$n≥5$$, $$A_n$$ is the only proper nontrivial normal subgroup of $$S_n$$.

For the proof, we firstly start with assuming a subgroup of $$S_n$$ which $$1≠N⊲S_n$$. We proceed until at the last part of proof's body, we assume $$N∩A_n=\{1\}$$. This assumption should be meet a contradiction with normality of $$N$$ in $$S_n$$. There; we get $$N=\{1,\pi$$} in which $$\pi$$ is an odd permutation of order $$2$$. Now for meeting desire inconsistency, I have two approaches:

(a) Since every normal subgroup, having two elements, lies in the center of $$G$$ so, our $$N⊆ Z(S_n)=\{1\}$$ for $$n≥5$$ and then $$N=\{1\}$$. (b) Clearly, $$1≠N$$ acting on set $$\Omega=\{1,2,...,n\}$$ is intransitive wherein $$|\Omega|≥5$$ and according to the following Proposition $$S_n$$ would be imprimitive. Proposition 7.1: If the transitive group $$G$$ contains an intransitive normal subgroup different from $$1$$, then $$G$$ is imprimitive (Finite Permutation Groups by H.Wielandt).

May I ask if the second approach is valid? I am fond of knowing new approach if exists. Thanks.

• Yes it is valid, but by the time you know |N|=2, you know N is central. Perhaps use the primitivity result earlier to conclude that N is transitive, and so N does intersect An. – Jack Schmidt Jun 18 '12 at 18:31
• @JackSchmidt: Thanks Jack. Honestly, I did the second one and wanted to give it to my Prof. :-) – mrs Jun 18 '12 at 18:39
• Make sure you can easily prove the result in Wielandt (it is easy, orbits of a normal subgroup are blocks), and it sounds fine. (a) is even easier though :-) – Jack Schmidt Jun 18 '12 at 18:41

You are almost there. Try to prove that $Z(S_n)= 1$ for all $n \geq 3$. Then if $N$ is non-trivial and normal, you assume $N \cap A_n = 1$. This implies $N \subseteq Z(S_n)$. Why? Because in general, if $N \unlhd G$ and $N \cap [G,G] = 1$ then $N \subseteq Z(G)$.
We conclude that the normal subgroup $N \cap A_n \neq 1$. At this point I assume that you know that $A_n$ is a simple group for $n \geq 5$. Hence $N \cap A_n = A_n$, so $A_n \subseteq N \subseteq S_n$. Since $index[S_n:A_n] = 2$, it follows that $N=A_n$ or $N=S_n$.
• @Babak, you are welcome! Your question got me thinking about the general case: can one classify all finite groups $G$ possessing a single proper non-trivial normal subgroup $N$? Here $N$ must be characteristic and characteristically simple and hence a direct product of isomorphic simple groups. Can more be said here about the structure of $N$ and $G$? Have to think about this. – Nicky Hekster Jun 18 '12 at 21:16