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I'm sure there must be lots of examples of three (infinite) groups $A$, $B$, $C$ where $A , $A\cong C$, $A\not\cong B$.

Here's one example with free groups: $\mathbf F(a^2,ab) < \mathbf F(a^2,ab,ab^{-1}) < \mathbf F(a,b)$, where A,C are a free group on 2 symbols and B is a free group on 3 symbols (in fact precisely the strings of even length in $\mathbf F(a,b)$.

But are there any examples that are just as easy but do not involve free groups (or equivalent constructions)?

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Take C to be the direct sum of countably many copies of Z/4Z, and A to be the direct summand where you omit one of the Z/4Z summands, call it X. So A⊕X = C and X≅Z/4Z. Then B=A⊕2X is not isomorphic to A or C, but A and C are isomorphic.

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    Ofcourse. That's a nice example, thanks!2011-01-05