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Given a group $G$ let $R(G)$ be its residual, that is the intersection of all the normal subgroups of finite index. Is there a name for the relation between $G$ and $H$ defined by $G/R(G) \cong H/R(H)$? (We thought of calling it finitely isomorphic.)

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Mark I don't understand your comment. $G$ and $H$ are two distinct groups, they are not subgroups of anything common. –  Yiftach Barnea Mar 30 '11 at 12:20
    
I see: I misread your question assuming that $H\le G$. –  Mark Sapir Mar 30 '11 at 12:38
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