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  1. The infinite dihedral group $D_\infty = \langle a,b \mid a^2 = b^2 = \text{Id}\rangle $ is a finitely generated group with infinitely many cyclic subgroups of order 2, every one of which is a retract.

  2. For the group $\mathbb{Z}\oplus\mathbb{Z}$‎, ‎take $H_n=\langle (1,n)\rangle$ for any integer $n$ (with $K=\langle (0,1)\rangle$)‎. ‎Then we have $\mathbb{Z}\oplus \mathbb{Z}=H_n \oplus K$ which shows that $\mathbb{Z}\oplus\mathbb{Z}$ (hence every finitely generated abelian group) has infinitely many different retracts‎.

  1. Every free group $F_n$ of finite rank $n$ has infinitely many retracts. In fact, each free factor of $F_n$ is a retract and there are infinitely many free factors.

These are examples of finitely generated groups with infinitely many retracts. If we look at them, we'll find that they have finitely many retracts up to isomorphism. My question is that does every finitely generated group have finitely many retracts up to isomorphism?

M.Ramana
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    You can add finitely generated free groups to your list of examples. Any retract of $F_n$ has rank $\leq n$. Maybe look for a counterexample in groups like Grigorchuk's Group. – Cheerful Parsnip Sep 11 '22 at 17:59
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    @CheerfulParsnip You are right. I'll add free groups of finite rank to my list. Thanks very much for the comment. – M.Ramana Sep 13 '22 at 12:43

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