Is there an isomorphism from G to (G/M)x(G/N) with the kernel M\bigcapN?

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I will try that.In summary, the conversation discusses how to prove that for normal subgroups M and N of group G, with G=MN, G/(M\bigcapN)\cong(G/M)x(G/N). The speaker has tried using a proposed isomorphism f(a) = (aM, aN) but is struggling to show it is onto. Another speaker suggests trying a = m' n, taking advantage of the normality of M and N.
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Let M and N be normal subgroups of G such that G=MN.
Prove that G/(M[itex]\bigcap[/itex]N)[itex]\cong[/itex](G/M)x(G/N).

I tried coming up with an isomorphism from G to (G/M)x(G/N) such that the kernel is M[itex]\bigcap[/itex]N, so that I can use the fundamental homomorphism theorem.
I tried f(a) = (aM, aN). It is an homomorphism and M[itex]\bigcap[/itex]N is the kernel but I'm having a hard time showing it is onto.

I would appreciate any help.
Thank you
 
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  • #2
So you need to show that for all g, g' in G, there is an a in G such that: (g M, g' N) = (a M, a N).

I haven't worked this out in detail, but: since G = MN, you can write g = m n, g' = m' n'. I suspect that a = m' n might do the trick.
You will need that M and N are normal, so in particular h M = M h, h N = N h for all h in G.
 
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Thank you Compuchip
 

1. What does it mean for two objects to be isomorphic?

Two objects are isomorphic if they have the same structure and can be mapped onto each other in a one-to-one correspondence. In other words, they are essentially the same object, just with different labels or representations.

2. How can you show that two objects are isomorphic?

To show that two objects are isomorphic, you must demonstrate that there exists a bijective mapping between them. This means that every element in one object corresponds to a unique element in the other object, and vice versa. This can be done by explicitly constructing the mapping or by proving that one already exists.

3. Are there different types of isomorphism?

Yes, there are different types of isomorphism depending on the type of objects being compared. For example, in mathematics, there are algebraic, geometric, and topological isomorphisms. In computer science, there are data structure and graph isomorphisms. Each type of isomorphism has its own specific requirements for showing that two objects are isomorphic.

4. Why is isomorphism important in science?

Isomorphism is important in science because it allows us to compare and understand complex systems by breaking them down into simpler, isomorphic components. It also helps us to identify patterns and relationships between seemingly different objects, leading to a deeper understanding of their underlying structure and behavior.

5. Can two objects be isomorphic but not identical?

Yes, two objects can be isomorphic but not identical. Isomorphism is a structural concept, so it focuses on the arrangement and relationships of the elements within an object rather than the individual elements themselves. This means that two objects can look different on the surface, but still have the same underlying structure and therefore be isomorphic.

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