# Classes of groups with finitely many retracts

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.

• I was going to mention simple groups as a trivial example.
– YCor
Commented Jun 6, 2023 at 17:07
• Prüfer $p$-groups are another example Commented Jun 6, 2023 at 17:17
• 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).
– YCor
Commented Jun 6, 2023 at 17:23
• 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? Commented Jun 6, 2023 at 19:40
• @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. Commented Jun 6, 2023 at 20:07

Any simple group $$G$$ only has $$G$$ and the trivial subgroup as retracts.
• 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. Commented Jun 7, 2023 at 5:21