Showing Subgroups of a Permutation Group are Isomorphic

In summary, permutation groups are mathematical structures that represent all possible arrangements or reorderings of a set of objects. If two subgroups of a permutation group are isomorphic, it means that they have the same structure and can be mapped onto each other in a one-to-one correspondence. To determine isomorphism, you can compare elements or check for the same order and cycle structure. Showing isomorphism between subgroups can simplify understanding and allow for generalizations and applications. There are techniques such as comparing elements, using generators, and examining structure, as well as specific theorems and properties that can be used to prove isomorphism between subgroups of a permutation group.
  • #1
Obraz35
31
0
Define two subgroups of S6:
G=[e, (123), (123)(456)]
H=[e, (14), (123)(456)]

Determine whether G and H are isomorphic.

It seems as if they should be since they have the same cardinality and you can certainly map the elements to one another, but I don't know what other factors need to be considered when deciding whether they are isomorphic.
 
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  • #2
H is not a subgroup since (14)(123)(456) = (123456) which is not in H. Are you sure you've written down the correct group?
 
  • #3
Oops. I meant G=<(123), (123)(456)> and H=<(14), (123)(456)>.
 

1. What are permutation groups?

Permutation groups are mathematical structures that represent all possible arrangements or reorderings of a set of objects.

2. What does it mean for two subgroups of a permutation group to be isomorphic?

If two subgroups of a permutation group are isomorphic, it means that they have the same structure and can be mapped onto each other in a one-to-one correspondence.

3. How can you determine if two subgroups of a permutation group are isomorphic?

To determine if two subgroups of a permutation group are isomorphic, you can compare their elements and see if there is a way to map one set of elements onto the other set. Alternatively, you can check if the subgroups have the same order and cycle structure.

4. Why is it important to show that subgroups of a permutation group are isomorphic?

Showcasing isomorphism between subgroups of a permutation group can help simplify the understanding of the group and its properties. It also allows for generalizations and application of results from one subgroup to the other.

5. Are there any techniques or methods for proving isomorphism between subgroups of a permutation group?

Yes, there are several techniques and methods for proving isomorphism between subgroups of a permutation group, including comparing elements, using generators, and examining the structure of the subgroups. Additionally, there are specific theorems and properties that can be applied to determine isomorphism.

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