# Abstract Algebra-Isomorphism

1. ### TheForumLord

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

2. ### ystael

352
Use the internal characterization of direct products of groups: if $$G$$ has two normal subgroups $$H, K$$ such that $$HK = G$$ and $$H \cap K = 1$$, then $$G \cong H \times K$$.

Also, the third isomorphism theorem may help you (if $$K \subset H$$ are both normal subgroups of $$G$$, then $$G/H \cong (G/K)/(H/K)$$).

3. ### TheForumLord

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

4. ### TheForumLord

108
I've managed to prove it...TNX a lot anyway...