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Gabrielle Horn

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So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct:

G/N=<(g1 * ... *gn)*k>

Where k is the generator of N and * the group operation. To create each coset g1N... gnN, we simply take gi for i=0,1,...n and then set all other (n-1) elements to the identity under the group operation, {e}. Thus we have n generators for g, but only one generator for G/N. Is this reasoning sound?