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  "body": "\n## Summary\n\nA [[group|G5S1M9]] is Hopfian if every surjective homomorphism from the group to itself is an isomorphism.\nEquivalently, the group is not isomorphic to any of its proper quotients.\nFinite groups, $\\Z$ and every finitely generated [[residually finite group|HEE922EHQ8]] are Hopfian.\nThe [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(2, 3)$ is a finitely presented group that is not Hopfian.\n\n## Explanations\n\n### L1 human\n\nA homomorphism can collapse parts of a group.\nA Hopfian group cannot collapse a part and still look like the whole group afterwards.\nFor finite groups this is clear by counting: a map from a finite set onto itself is a bijection.\nAn infinite group can fail.\nTake the group of all sequences $(x_1, x_2, \\ldots)$ of elements of $\\Z/2$ that are $0$ from some point on.\nDeleting the first term is onto, but it kills the sequence $(1, 0, 0, \\ldots)$.\n",
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