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Suppose we have a triple of groups $G,H,K$ satisfying the following conditions:

  1. $G$ and $H$ are finite groups and $K$ is an infinite group.
  2. there exist two monomorphisms $G \rightarrow K \leftarrow H$ which induce an isomorphism in homology (with integral coefficients).

My question is the following: Is there a known triple $(K, G, H)$ satisfying the conditions (1.) and (2.) such that $G$ is not isomorphic to $H$? I am more interested in the case that $G$ is a perfect group, but any example (if there exists such) is welcome.

Edit: In the case where $K$ is finite, Culler's theorem says that $K, G, H$ are all isomorphic.

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  • $\begingroup$ Does the proof of the case $K$ being finite has anything to do with some Eilenberg-Moore type spectral sequence? And how a comparison map $G\to H$ is constructed in the finite case? Is it constructed geometrically? $\endgroup$
    – user51223
    Commented May 14, 2016 at 7:27
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    $\begingroup$ @user51223 Here is the reference jstor.org/stable/2042568?seq=1#page_scan_tab_contents $\endgroup$
    – Ilias A.
    Commented May 16, 2016 at 18:14
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    $\begingroup$ @IliasA. I have undeleted this question which you have deleted last year. -- Please do not self-delete your useful questions! $\endgroup$
    – Stefan Kohl ♦
    Commented Apr 30, 2021 at 20:55
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    $\begingroup$ Do you have an example of an infinite group K and a finite subgroup G such that the inclusion induces an isomorphism in cohomology? $\endgroup$
    – Todd Leason
    Commented May 11, 2021 at 12:18

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