Abstract
Solving equations belongs to the oldest problems in mathematics. In a wider sense, analyzing a group presentation belongs to this class of problems, because one wants to know the most general solutions of the defining relations. It is well known that there is no general procedure to solve the word problem by the famous Novikov-Boone Theorem, cf. [30] Chapter 13 for an exposition. Even deciding the question whether G is finite or infinite cannot be solved in general. Nevertheless, one can try to prove that G is infinite, if one suspects this, by solving the equations given by the relators in some group, where one can compute, for example in a matrix group. In case of success this produces an epimorphic image of G which might be infinite. And even if it is finite, one might use the representations of the finite quotient to produce bigger epimorphic images. Various techniques for carrying out these ideas have been developed over the last years. They will be described in the next few chapters.
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Nebe, G. (1999). The Structure of Maximal Finite Primitive Matrix Groups. In: Matzat, B.H., Greuel, GM., Hiss, G. (eds) Algorithmic Algebra and Number Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59932-3_21
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DOI: https://doi.org/10.1007/978-3-642-59932-3_21
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