Hopfianness is not subgroup-closed

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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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Statement

It is possible to have a Hopfian group G and a subgroup H of G that is not Hopfian.

Related facts

Stronger facts

Opposite facts

Proof

Let G be free group:F2 and H be the derived subgroup of G.

  • G is Hopfian because finitely generated and free implies Hopfian.
  • H is not Hopfian: H 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 H to itself.