Retraction Math Help: Does a Group Exist with Trivial Retracts?

[AKG]

Any direct sum of denumerably many groups will have (at least) denumerably many retracts, and any direct product of denumerably many groups will have uncountably many retracts. The projection maps are retractions.

 

[Hurkyl]

Any direct sum of denumerably many groups will have (at least) denumerably many retracts

That "at least" is worrying me. Any countably infinite direct sum of (nontrivial) groups has uncountably many retracts; one for every subset of the index set. (and possibly more) I suspect that an abelian group cannot have exactly countably infinite retracts... and if I'm wrong, it will still be a tricky business finding an example.

 

[AKG]

That "at least" is worrying me. Any countably infinite direct sum of (nontrivial) groups has uncountably many retracts; one for every subset of the index set.

Yeah, you're right.