I'm trying to find a characterisation of all normal endomorphisms on the symmetric group $S_n$ (and the alternating group $A_n$). I know that the image of a normal subgroup under a normal endomorphism is again normal, but that is all. I don't know where to start. Any help or solution is appreciated.
Thanks for the comments. I am trying to get a bettter picture of the situation.
OK, so for $n\geq 5$, $S_n$ only has three normal subgroups, $S_n$ itself, $A_n$ and the trivial subgroup. That would mean the only possible endomorphisms are the trivial endomorphism mapping everything to the identity permutation (which of course is a normal endomorphism), some kind of automorphism or an endomorphism mapping $S_n$ surjectively onto $A_n$.
The last case is not possible, I think, due to the fact that the kernel of such a homomorphism would then be a normal subgroup of order 2, of which there is none in $S_n$.
Any help on the automorphisms?