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Complete Set Of Relations

  • Definition: Given G, suppose we are given a family of generators {x_a}_(a in J) of generators for G. Let F be the the free group on the elements {x_a}. Then the obvious map h(x_a) = x_a of these elements into G extends to a homomorphism h: F -> G that is surjective. If N equals the kernel of h, then F/N is isomorphic to G. So one way of specifying G is to give a family {x_a} of generators for G, and somehow to specify the subgroup N. Each element of N is called a relation on F, and N is called the relations subgroup. We can specify N by giving a set of generators for N. But since N is normal in F, we can also specify N by a smaller set. Specifically, we can specify N by giving a family {r_j} of elements of F such that these elements and their conjugates generate N, that is, such that N is the least normal subgroup of F that contains the elements r_j. The family {r_j} is called a complete set of relations for G.

    Source: Topology (second edition) by James R. Munkres

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