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Let $G$ be a group. A subgroup $H$ of $G$ is called a retract of $G$ if there exists a homomorphism $r:G \to H$ such that $r(h)=h$, for all $h\in H$.

We can easily check that every finite-by-cyclic group has only finitely many retracts (as a subgroup). (Recall that a group $G$ is finite-by-cyclic if it is an extension of a finite group by a cyclic group).

My question: Is there a bigger class (or other class) of groups with finitely many retracts?

Sorry, I edit my question. Trivially, every simple group (even more splitting groups: groups with no proper nontrivial retracts) are triviall examples. I am looking for other examples.

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  • $\begingroup$ I was going to mention simple groups as a trivial example. $\endgroup$
    – YCor
    Jun 6, 2023 at 17:07
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    $\begingroup$ Prüfer $p$-groups are another example $\endgroup$
    – Emil Jeřábek
    Jun 6, 2023 at 17:17
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    $\begingroup$ And more generally, indecomposable abelian groups (this includes all subgroups of $\mathbf{Q}$, but larger ones, including "most" subgroups of $\mathbf{Q}^n$, and also, for an uncountable example, the group of $p$-adics). $\endgroup$
    – YCor
    Jun 6, 2023 at 17:23
  • $\begingroup$ The usual usage is that a group is X-by-Y if it has a normal subgroup in class X with quotient in class Y. This is also the usual meaning of an extension of a group in class Y by a group in class X, NOT of one in class X by one in class Y. So which do you mean? $\endgroup$
    – Dave Benson
    Jun 6, 2023 at 19:40
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    $\begingroup$ @DaveBenson In my experience "extension of group $X$ by group $Y$" is used about equally often with both meanings, so there is no usual meaning. In any case the "Recall that ..." sentence in the post deos not resolve the possible confusion. $\endgroup$
    – Derek Holt
    Jun 6, 2023 at 20:07

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Any simple group $G$ only has $G$ and the trivial subgroup as retracts.

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  • $\begingroup$ Sorry, I forgot to say that that I'm looking for groups with finitely many retracts other simple groups because they are trivial examples with just two retracts. Even more, splitting-simple groups have only two retracts like $\mathbb{Z}$ and every simple group is splitting-simple. By a splitting-simple group, I mean a group which has no proper nontrivial retracts. $\endgroup$
    – M.Ramana
    Jun 7, 2023 at 5:21

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