Isomorphism of Quotient Groups

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    Abstract
In summary, the problem is asking to prove that G/AnB is isomorphic to G/A*G/B, and the solution involves using the internal characterization of direct products of groups and possibly the third isomorphism theorem. The conversation also discusses the use of latex text and clarifies the conditions needed for the internal characterization to be applicable. The person eventually figures out how to solve the problem.
  • #1
TheForumLord
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Homework Statement



Let A,B be normal sub-groups of a group G.
G=AB.

Prove that:
G/AnB is isomorphic to G/A*G/B

Have no idea how to start...Maybe the second isom. theorem can help us...

TNX!



Homework Equations


The Attempt at a Solution

 
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  • #2
Use the internal characterization of direct products of groups: if [tex]G[/tex] has two normal subgroups [tex]H, K[/tex] such that [tex]HK = G[/tex] and [tex]H \cap K = 1[/tex], then [tex]G \cong H \times K[/tex].

Also, the third isomorphism theorem may help you (if [tex]K \subset H[/tex] are both normal subgroups of [tex]G[/tex], then [tex]G/H \cong (G/K)/(H/K)[/tex]).
 
  • #3
Sry but I rly can't figure out the Latex text (I see it in black, and it's really not clear)...
If I understand what you're saying, then we don't have the right conditions to use "internal characterization of direct products of groups"...
A,B are normal sub-groups of G and AB=G but who said AnB={1}? The isomorphism you've put afterwards is relevant only when G=A*B and it isn't the case///

Am I wrong?

TNx
 
  • #4
I've managed to prove it...TNX a lot anyway...
 

Related to Isomorphism of Quotient Groups

1. What is an isomorphism in abstract algebra?

An isomorphism is a type of function that preserves the structure and operations of mathematical objects, such as groups, rings, and fields. It is a one-to-one and onto mapping between two algebraic structures that preserves the properties and relationships between their elements.

2. How is isomorphism different from homomorphism?

Homomorphism is a more general type of function that preserves the operations of mathematical structures, but not necessarily the structure itself. Isomorphism is a stricter form of homomorphism in which the function preserves both the operations and the structure of the objects.

3. Can two isomorphic algebraic structures be considered the same?

Yes, two isomorphic structures can be considered equivalent in the sense that they have the same abstract structure and operations, even if the elements themselves are different. For example, two groups may have different elements, but if they are isomorphic, they have the same group structure and operations.

4. How can you prove that two algebraic structures are isomorphic?

To prove that two algebraic structures are isomorphic, you must show that there exists a bijective homomorphism between them. This can be done by explicitly constructing the function and showing that it preserves the operations and structure of the objects, or by demonstrating that the structures have the same properties and can be mapped onto each other.

5. What is the significance of isomorphism in abstract algebra?

Isomorphism is a powerful tool in abstract algebra that allows us to study and understand complex mathematical structures by relating them to simpler or more familiar ones. It also helps us to identify important properties and relationships between different algebraic structures, and can be used to prove theorems and solve problems in various areas of mathematics.

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