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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\n$G$ is *Hopfian* if every [[surjective|01RVCMSN5N]] group [[homomorphism|0JYTG5K9Y4]] $\\varphi \\colon G \\to G$ is [[injective|09NWETVTNY]].\n\n## Remarks\n\nEquivalently, if $N$ is a normal subgroup of $G$ and $G/N \\cong G$, then $N = \\set{1}$.\n\nExample: $\\Z$ is Hopfian.\nA homomorphism $\\varphi \\colon \\Z \\to \\Z$ is $n \\mapsto kn$ with $k = \\varphi(1)$.\nIf it is surjective, then $k = \\pm 1$, and then it is injective.\n\nNon-example: let $G = \\bigoplus_{n \\ge 1} \\Z/2$, the group of sequences $(x_1, x_2, \\ldots)$ in $\\Z/2$ with only finitely many nonzero terms.\nThe shift $\\varphi(x_1, x_2, x_3, \\ldots) = (x_2, x_3, \\ldots)$ is a surjective homomorphism.\nIt is not injective, since $\\varphi(1, 0, 0, \\ldots) = 0$.\nA finitely generated non-Hopfian group is the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(2, 3)$.\n\nThe dual notion is a co-Hopfian group: every injective homomorphism from the group to itself is surjective.\n",
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