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

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A subgroup H of a group G is defined as a retract if there exists a homomorphism from G to H that fixes every element of H. The discussion explores the possibility of forming a subgroup K, which is the kernel of this retraction, and examines the conditions under which G can have only trivial retracts. It concludes that groups like Z4 and Z/p^n for primes p and n > 1 are non-simple groups with trivial retracts. The conversation also highlights that the existence of non-trivial retracts is linked to whether a group can be expressed as a direct product of other groups. Ultimately, the thread seeks to identify groups that meet specific criteria regarding retracts and normal subgroups.
  • #31
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.
 
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  • #32
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.
 
  • #33
Hurkyl said:
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.
 

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