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# Conjugacy Classes watch

1. Hey guys, really stuck on this question:

Let
Let H = {1,(12)(34)}
Let J = {1, (12)(34),(13)(24),(14)(23)}

Show that:

but

I understand J consists of the conjugacy class cycles of type (2,2) in S4, but how do I deduce from this that and then ? Is it true in general if then ?
2. Well if and only if for all , so if then since for all , we must also have .

To show that you need to use the fact if and then ; that is, conjugation preserves cycle type.
3. J also consists of the conjugacy classes of type (2,2) in A4 (since all (2,2) types are in A4), so I think you can just work directly in A4.

If g has a certain cycle structure then obviously will have the same structure, so any (2,2) cycle will remain a (2,2) cycle under conjugacy. So for all j in J and so J is normal in G.
4. (Original post by nuodai)
Well if and only if for all , so if then since for all , we must also have .

To show that you need to use the fact if and then ; that is, conjugation preserves cycle type.

Thank you! It makes sense to me now that showing .

Can you say: "J contains all the possibilities of cycle types (2,2), and therefore (as well as J < S4) conjugating any element of J will also remain in J (because we've got all the elements of type (2,2) ) so J is normal in S4 (and A4)."

Thank you both for your help!
5. (Original post by sammmmmm)
Thank you! It makes sense to me now that showing .

Can you say: "J contains all the possibilities of cycle types (2,2), and therefore (as well as J < S4) conjugating any element of J will also remain in J (because we've got all the elements of type (2,2) ) so J is normal in S4 (and A4)."
Yup, although you might want to make it a bit clearer by writing out a few more details.
6. (Original post by nuodai)
Yup, although you might want to make it a bit clearer by writing out a few more details.
Thank you very much for your help!

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