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Abstract Algebra-Isomorphism

by TheForumLord
Tags: abstract, algebraisomorphism
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TheForumLord
#1
Dec18-09, 08:29 AM
P: 108
1. The problem statement, all variables and given/known data

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!



2. Relevant equations
3. The attempt at a solution
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ystael
#2
Dec18-09, 12:08 PM
P: 352
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]).
TheForumLord
#3
Dec18-09, 12:38 PM
P: 108
Sry but I rly can't figure out the Latex text (I see it in black, and it's realy 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

TheForumLord
#4
Dec18-09, 03:09 PM
P: 108
Abstract Algebra-Isomorphism

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


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