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Let $G$ be an abelian group and let $A$ and $B$ be subgroups of $G$. Furthermore, let $C$ be a subgroup of $A \cap B$. I would like to find another subgroup $A+B \subseteq D \subseteq G$ so that $D/(A+B) \cong (A \cap B)/C$. In general I'm not sure this is possible although in the situation I am really interested in I am pretty sure it is true. Therefore, my questions are:

1) Are there additional restrictions that I can place on the situation to guarantee the existence of such a subgroup $D$?

2) If a $D$ exists and I know a set of representatives for $(A \cap B)/C$ is it possible to deduce a set of representatives for $D/(A+B)$.

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If $A+B=G$, then the only option for $D$ is $G$. So you want in that case $G/(A+B)=1=(A\cap B)/C$. This can only happen when $C=A\cap B$. This rarely happens if $A\cap B$ is large. Perhaps some conditions are missing. On the other hand, if $A\cap B=1$, then you can take $D=A+B$. –  Mark Sapir Feb 23 '13 at 15:18
    
The only case that need to be considered is that $C=1$. And now the equation is $D/(A+B) \cong A \cap B$. In general, this is not true. So there need some additional restrictions. –  Wei Zhou Feb 24 '13 at 0:36
    
There are certainly several cases of my question that are uninteresting. In the case I actually am interested in, A+B is a proper subgroup of G and C is smaller than $A \cap B$ but still bigger than just $1$. –  user4535 Feb 24 '13 at 4:15

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