Cyclic Quotient Group: Is My Reasoning Sound?

[Gabrielle Horn]

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?

 

[Erland]

So it's apparent that G/N cyclic --> G cyclic.

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

Perhaps you meant it the other way round: G cyclic --> G/N cyclic?