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Cyclic quotient group?

  1. Oct 27, 2015 #1
    Hi everyone.

    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?
  2. jcsd
  3. Oct 28, 2015 #2


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    But that's not true. Zp + Zp (direct sum, p prime) is not cyclic, but (Zp + Zp) / Zp (with one of the summands as the denominator) is isomorphic to Zp, hence cyclic.

    Perhaps you meant it the other way round: G cyclic --> G/N cyclic?
    Last edited: Oct 28, 2015
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