Isomorphism of Quotient Groups

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Homework Help Overview

The discussion revolves around proving an isomorphism between quotient groups involving normal subgroups of a group G. The original poster presents a problem statement regarding the relationship between G, its normal subgroups A and B, and their intersection.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the applicability of the second and third isomorphism theorems, with some questioning the conditions necessary for these theorems to hold. There is a discussion about the internal characterization of direct products and the implications of the intersection of subgroups.

Discussion Status

The discussion includes attempts to clarify the conditions under which certain theorems can be applied. Some participants express uncertainty about the assumptions regarding the intersection of the subgroups, while others provide insights into potential approaches. The original poster indicates a resolution to their problem, suggesting that some productive direction was achieved.

Contextual Notes

There is a noted concern regarding the conditions for applying the internal characterization of direct products, particularly the assumption that the intersection of the subgroups is trivial. This highlights a potential gap in the information provided in the problem statement.

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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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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]).
 
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
 
I've managed to prove it...TNX a lot anyway...
 

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