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von Dyck's Theorem

Let a Group $G$ have a presentation

G=(x_1,\ldots,x_n\vert r_j(x_1,\ldots,x_n), j\in J)

so that $G=F/R$, where $F$ is the Free Group with basis $\{x_1, \ldots, x_n\}$ and $R$ is the Normal Subgroup generated by the $r_j$. If $H$ is a Group with $H=\left\langle{y_1,\ldots,y_n}\right\rangle{}$ and if $r_j(y_1,\ldots,y_n)=1$ for all $j$, then there is a surjective homomorphism $G\to H$ with $x_i\mapsto y_i$ for all $i$.

See also Free Group, Normal Subgroup


Rotman, J. J. An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, p. 346, 1995.

© 1996-9 Eric W. Weisstein