Hopfianness is not subgroup-closed
This article gives the statement, and possibly proof, of a group property (i.e., Hopfian group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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- is Hopfian because finitely generated and free implies Hopfian.
- is not Hopfian: is a free group on a countable number of generators. It admits a surjective endomorphism that is not an automorphism obtained via a surjective map that is not bijective from the freely generating set for to itself.