# Finding Subgroups of certain orders

I'm not sure how for example I would go about finding the subgroups of S_4 of order 4?

I don't quite understand how I would I find or construct subgroups for a permutation group?

Thanks

lavinia
Gold Member
I'm not sure how for example I would go about finding the subgroups of S_4 of order 4?

I don't quite understand how I would I find or construct subgroups for a permutation group?

Thanks
1 ->2 ->3 ->4 ->1
1 ->3 ->2 ->4 ->1 for example

jgens
Gold Member
Suppose $\sigma \in S_n$ is a 4-cycle. What can you say about $\langle \sigma \rangle$?

I used (1 2 3 4) and generated a subgroup of order 4, this subgroup was, {id, (1234), (13)(24), (1432)}, is this correct? And this is of order 4 so meets the condition.
I also did the same with (1324).

After reading a theorem I concluded if I want the subgroups to be of order 4 then the elements of the subgroups must divide 4. So since I've done the two for 4 would I know need to do the same for elements of order 2? Is there a shortcut or do I have to compute them all to check they form subgroups of order 4?

After attempting this with for example (12)(34) I found the usual method of multiplying by itself to try and find a group quickly failed since (12)(34)(12)(34) is simply id.

How do I find the subgroups using elements of order 2?