How do I find an isomorphism between Sn+m and Zn x Zm?

In summary, the conversation discusses finding an isomorphism between Sn+m and Zn x Zm, where n and m are not relatively prime. The conversation suggests placing an ad, defining the notation, and following guidelines for posting in the homework section. It also mentions finding an isomorphism between a subgroup in Sn+m and Zn x Zm, with specific cases such as n,m=2,2 being easily solvable. Lastly, the conversation mentions that if two subgroups of a group have a trivial intersection and satisfy hk=kh, then the subgroup HK is isomorphic to HxK.
  • #1
clkt
6
0
How do I find an isomorphism between Sn+m and Zn x Zm? provided n,m are not relatively prime? Thanks.
 
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  • #2
I dunno, place an ad in the classifieds, put up posters, etc. Or maybe start by defining what you even mean by Sn+m, because in the standard notation, Sn+m is a group that's generally not isomorphic to Zn x Zm. Also, post this in the homework section, follow the guidelines for posting homework, and when you start a new thread in the homework section, use the template that will be given to you.
 
  • #3
Hi, this isn't exactly a homework question... Anyways, rereading my question again, I made a mistake. It should be how do I find an isomorphism between A SUBGROUP in Sn+m and Zn x Zm. If it is specific cases like n,m = 2,2 I can figure it out. But how do I figure out a general form for the isomorphism?
 
  • #4
If there are two subgroups H and K of a group G such that the intersection of H and K is trivial and hk=kh for all h in H, k in K, then the subgroup HK is isomorphic to HxK.
 
  • #5
ah, thanks, I think I have it figured out now!
 

1. What is an isomorphism?

An isomorphism is a type of mathematical function that preserves the structure and relationships between elements of two different mathematical objects. In simpler terms, it is a one-to-one mapping between two mathematical structures that preserves their properties.

2. How can I find an isomorphism between Sn+m and Zn x Zm?

To find an isomorphism between these two structures, you will need to show that there is a one-to-one mapping between the elements of Sn+m and Zn x Zm that preserves their properties. This can be done by finding a function that maps each element of Sn+m to a unique element in Zn x Zm, and vice versa, while also preserving the group operation and other properties.

3. What are the properties that need to be preserved in an isomorphism?

In order for an isomorphism to exist between two mathematical structures, certain properties must be preserved. These include the group operation, which must remain the same under the mapping, and the identity element, which must be mapped to the identity element. Other properties such as inverses and commutativity must also be preserved.

4. Is there a unique isomorphism between Sn+m and Zn x Zm?

Yes, there is a unique isomorphism between these two structures. This is because once an isomorphism is established, it is unique and cannot be changed without violating the properties that need to be preserved. Therefore, there can only be one function that serves as an isomorphism between Sn+m and Zn x Zm.

5. Can an isomorphism exist between any two mathematical structures?

No, not all mathematical structures can have an isomorphism between them. For an isomorphism to exist, the structures must have similar properties and relationships between elements. Additionally, the structures must be of the same type, such as groups, rings, or fields, for an isomorphism to be possible.

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