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Is there a list of all normal subgroups for $S_N$?

What is a criteria for a finite group to be a normal subgroup of $S_N$?

Which of them are kernels of irreducible representation? From a partition of $N$, we can construct an irreducible representation, so how does the related subgroup look in terms of the partition?

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up vote 9 down vote accepted

It is a standard fact in group theory that when $N\geq 5$, the only normal subgroup is $A_N$, which in turn is simple. For $N\leq 4$, you can just do it by hand. There are is of course exactly two one irreducible representation whose kernel is $A_N$, the trivial and the sign representation.

I don't understand the last paragraph of your question, but there are lots of references on representations of symmetric groups, e.g. the representation theory book by Fulton and Harris.

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I'd say there's one irreducible representation whose kernel is $A_N$, and one whose kernel contains $A_N$ (but is $S_N$). – Gerry Myerson Jun 9 '11 at 12:20
@Gerry: Of course. Thanks! – Alex B. Jun 9 '11 at 13:10
Thanks for this asnwer, so all other higher dimensional representation are faithful? – Jun 9 '11 at 17:56

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