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  "body": "\n## Statement\n\nLet $G = \\langle S \\mid R \\rangle$, let $H$ be a [[group|G5S1M9]], and let $f \\colon S \\to H$ be a function.\nLet $\\varphi \\colon F_S \\to H$ be the homomorphism with $\\varphi(s) = f(s)$ for each $s \\in S$.\nIf $\\varphi(r) = e$ for each $r \\in R$, then there is exactly one [[group homomorphism|0JYTG5K9Y4]] $\\psi \\colon G \\to H$ with $\\psi(s N) = f(s)$ for each $s \\in S$.\n\n## Remarks\n\nHere $N$ is the normal closure of $R$ in $F_S$, so $G = F_S / N$.\nThe proof combines the [[universal property of the free group|0CDWKDYWN6]] with the [[universal property of the quotient group|0GJ83E78A5]], because $N \\subseteq \\ker \\varphi$.\nIf the elements $f(s)$ generate $H$, then $\\psi$ is surjective, so $H$ is a quotient of $G$.\nThe name refers to Walther von Dyck, who gave the first systematic study of group presentations in the early 1880s.\n\n## Explanations\n\n### L1 human\n\nTo map a presented group to another group, choose the images of the generators.\nIf these images obey all the relations, then the choice gives exactly one homomorphism.\nYou do not need to check anything else.\n",
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