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What can be said about pairs of non-isomorphic groups which are epimorphic images of one another and which also embed into one another?

Can such pairs of groups be 'classified' in some sufficiently weak, but still non-trivial sense, or are they just too common to hope for anything like this?

Obviously, groups forming such pairs can neither be Hopfian nor co-Hopfian.

An example of such pair of groups consists of $\rm C_\infty \times \rm F_2^\infty$ and $\rm F_2^\infty$, where $\rm C_\infty$ denotes the infinite cyclic group and $\rm F_2$ denotes the (nonabelian) free group of rank 2.

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    $\begingroup$ A slightly simpler example: $C_2 \times C_4^\infty$ and $C_4^\infty$. $\endgroup$
    – S. Carnahan ♦
    Commented Jan 18, 2013 at 14:53
  • $\begingroup$ Indeed. -- However which one is 'simpler' depends on your notion of 'simplicity': in order to define a free group, you need no relations, while to define a finite cyclic group, you need 1! $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 18, 2013 at 15:00
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    $\begingroup$ The examples given so far are of the form $(A \times B^\infty, B^\infty)$, where $A$ both embeds into $B$ and is an epimorphic image of $B$. -- Can anyone give an example where the groups do not admit a decomposition into infinitely many direct factors, or which are at least not of the form $(A \times B^\infty, B^\infty)$? $\endgroup$
    – Stefan Kohl ♦
    Commented Jan 22, 2013 at 16:00
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    $\begingroup$ I believe you can modify the example so that the direct products become semidirect products. $\endgroup$
    – Khalid Bou-Rabee
    Commented Jan 28, 2013 at 5:03
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    $\begingroup$ ... and one could also take (so-called) direct sums of groups, add boring factors, etc. $\endgroup$
    – Martin Brandenburg
    Commented Oct 9, 2014 at 21:53

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