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von Dyck's Theorem


Let a group G have a group presentation

 G=<x_1,...,x_n|r_j(x_1,...,x_n),j in J>

so that G=F/R, where F is the free group with basis {x_1,...,x_n} and R is the normal subgroup generated by the r_j. If H is a group with H=<y_1,...,y_n> and if r_j(y_1,...,y_n)=1 for all j, then there is a surjective homomorphism G->H with x_i|->y_i for all i.


See also

Dyck's Theorem, Free Group, Normal Subgroup

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References

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

Referenced on Wolfram|Alpha

von Dyck's Theorem

Cite this as:

Weisstein, Eric W. "von Dyck's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/vonDycksTheorem.html

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