Finding Subgroups of certain orders

[RVP91]

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?

Any advice?

Thanks

 

[lavinia]

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?

Any advice?

Thanks

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

 

[jgens]

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

 

[RVP91]

After using both your advice I think I have made some progress.

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?

Thanks for the previous advice and an advance thanks for anymore

 

[CellCoree]

for your question! Finding subgroups of a certain order in a permutation group, such as S_4, can be a challenging task. However, there are some strategies and techniques that can help us in this process.

One approach is to use the concept of cyclic subgroups. In a permutation group, a cyclic subgroup is a subgroup generated by a single element. For example, in S_4, the cyclic subgroup generated by (1 2 3) would have order 3. By finding all possible cyclic subgroups of different orders, we can then combine them to form subgroups of the desired order.

Another method is to use the concept of cosets. In a permutation group, a coset is a set of elements that are obtained by multiplying a fixed subgroup by a single element. By finding all possible cosets of a given subgroup, we can then check which ones have the desired order.

Additionally, there are some specific techniques for finding subgroups in permutation groups, such as the Orbit-Stabilizer Theorem and the Sylow Theorems. These theorems provide powerful tools for identifying and constructing subgroups of a certain order.

In terms of constructing subgroups, it can also be helpful to think about the structure of the permutation group and its elements. For example, in S_4, we know that the elements are permutations of 4 objects, so we can try to construct subgroups based on the different ways these elements can be combined and manipulated.

Overall, finding subgroups of a certain order in a permutation group requires a combination of techniques and strategies. It can be a challenging task, but with patience and perseverance, we can identify and construct these subgroups. I hope this helps and good luck in your exploration!