Finding Subgroups of certain orders

• RVP91
In summary, the conversation is about finding the subgroups of a permutation group, specifically S_4 of order 4. The individual is seeking advice on how to construct these subgroups and has made progress by using a 4-cycle and generating a subgroup of order 4. They have also realized that for subgroups of order 4, the elements must divide 4 and are now wondering about finding subgroups using elements of order 2. They express gratitude for previous advice and ask for any further assistance.
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?

Thanks

RVP91 said:
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

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?

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!

1. What is the significance of finding subgroups of certain orders?

Finding subgroups of certain orders is important because it helps us understand the structure and properties of a larger group. By studying the subgroups, we can gain insight into the behavior and relationships within the group, and potentially make predictions or solve problems related to the group.

2. How do you determine the orders of subgroups?

The order of a subgroup is equal to the number of elements in the subgroup. To determine the order, we can count the number of elements in the subgroup or use mathematical formulas and techniques specific to the type of group being studied.

3. Can subgroups of certain orders have different properties from the larger group?

Yes, subgroups of certain orders can have different properties from the larger group. This is because subgroups are smaller, self-contained groups within the larger group, and may have their own unique characteristics and relationships between elements.

4. How are subgroups of certain orders related to other mathematical concepts?

Subgroups of certain orders are closely related to other mathematical concepts such as group theory, abstract algebra, and combinatorics. These concepts help us understand the structure and properties of subgroups and how they relate to other elements in the group.

5. Are there any practical applications of finding subgroups of certain orders?

Yes, there are many practical applications of finding subgroups of certain orders. For example, in cryptography, subgroups are used to create secure encryption methods. In chemistry, subgroups can be used to study molecular structures and reactions. In computer science, subgroups are utilized in data analysis and clustering techniques.

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