How Can I Identify Semipermutable Subgroups in S4?

[DoMutasem]

How i can find the semipermutable subgroups in S4?

i konw that the normal subgroup is semipermutable .

 

[Poopsilon]

Well S4 has 4! elements, thus any of its subgroups must have order which divide 4!, so we have possible orders 12, 8, 6, 4, 3, 2, 1. I think there's a theorem about subgroups of half the order being normal. I'm not sure if you've done any classifications yet, but there aren't that many groups of order less than 10, you could probably just check each one for being semipermutable.

 

[DoMutasem]

I had find all subgroups of S4,it is 30 subgroups, and i know the normal subgroup in it.
Normal is A4 and order 4.
But i want to check the rest,,how i can check, ?
and now i should depend on some thing else not normal, the normal role is finished...

i want to learn how to check the semipermubality ??

 

[Poopsilon]

30 subgroups! That is definitely not correct. The website groupprops.subwiki.org lists 11 subgroups including the trivial group and S4 itself.

I've actually never dealt with the concept of semipermutability myself, the method to check if a subgroup satisfies this property hopefully follows directly from its definition. But we know S4, A4, and the trivial subgroup are all normal in S4. So that leaves you with 8 more to check, so maybe reread the definition of semipermutability and see if you can devise a way to check the others.

 

[DoMutasem]

Thank you.
But the subgroups is 3o subgroup with 5 conjugacy class, http://groupprops.subwiki.org/wiki/Symmetric_group:S4

I m trying to check it one by one but it will take time,
Do you know the GAP program i had tried to download but i could not any idea,

Thank you

 

[Poopsilon]

I'm not sure what "Arithmetic Functions of a Counting Nature" exactly means, but I'm pretty sure that is not referring to the number of subgroups of S4, and if it is, then it's wrong, scroll down further and you will see a list of all 11 of them by name.

I don't know the GAP program.

 

[DoMutasem]

okay, Thank you,
i will do my calculation again