Subgroups of External Direct Products

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In summary, the conversation involves finding all subgroups of order 3 in the external direct product of Z9 and Z3. The elements and subgroups are listed, but there is a mistake in the subgroups as they are actually from Z3 x Z9, not Z9 x Z3. The error is acknowledged and the solution is to fix the subgroups and re-write the page. The conversation ends with gratitude for the help.
  • #1
moo5003
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Problem "Find all subgroups of order 3 in Z9 x Z3"

Using an external direct product I came out with the elements:

0,0
0,1
0,2
0... to 0,9
1,0
1... to 1,9
2,0
2... to 2,9


With subgroups:

{(0,0),(1,0),(2,0)}
{(0,0),(0,3),(0,6)}
{(0,0),(1,3),(2,6)}
{(0,0),(1,6),(2,3)}

Just looking for some confirmation if I have done this correctly. I was reading my book and it was lacking a definition/example that I felt like I fully understood.
 
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  • #2
This is 99% correct, it's just that you've given subgroups of Z3 x Z9, not Z9 x Z3.
 
  • #3
Wow... nice catch. Now I'm wondering if I can just write on top of the page to switch everything to the other side or if I should re-write the page :(.

I'll just write a little snippet about it and then re-write the subgroups. That seems to the best way about fixing this. Thanks for the help.
 

FAQ: Subgroups of External Direct Products

1. What is an external direct product?

An external direct product is a mathematical operation that combines two or more groups into a new group. It is denoted by the symbol × and is used to describe the product of two or more groups that are not necessarily subgroups of each other.

2. How do subgroups of external direct products differ from subgroups of normal direct products?

In subgroups of external direct products, the elements do not have to commute with each other, whereas in subgroups of normal direct products, the elements must commute with each other. This means that the order in which the elements are multiplied can affect the final result in subgroups of external direct products.

3. Can a subgroup of an external direct product be isomorphic to the direct product of subgroups?

Yes, it is possible for a subgroup of an external direct product to be isomorphic to the direct product of subgroups. This can occur when the subgroups are "compatible" with each other, meaning that their elements commute with each other in a specific way.

4. What is the significance of subgroups of external direct products in group theory?

Subgroups of external direct products are important in group theory because they allow for the creation of new groups from existing groups. This can help to simplify and understand the structure of more complex groups. Additionally, studying subgroups of external direct products can provide insights into the structure and properties of the original groups.

5. Are all subgroups of external direct products also subgroups of the individual groups?

No, not all subgroups of external direct products are also subgroups of the individual groups. This is because the elements in subgroups of external direct products do not necessarily commute with each other, so they may not be contained within the individual groups. However, some subgroups of external direct products may be subgroups of the individual groups if the elements commute with each other in a specific way.

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