6
$\begingroup$

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?

$\endgroup$
10
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ I'd say there's one irreducible representation whose kernel is $A_N$, and one whose kernel contains $A_N$ (but is $S_N$). $\endgroup$ – Gerry Myerson Jun 9 '11 at 12:20
  • $\begingroup$ @Gerry: Of course. Thanks! $\endgroup$ – Alex B. Jun 9 '11 at 13:10
  • $\begingroup$ Thanks for this asnwer, so all other higher dimensional representation are faithful? $\endgroup$ – Marc Palm Jun 9 '11 at 17:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.