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Factor Groups of Non-Abelian Groups

Let G be a nonabelian group and Z(G) be its center. Show that the factor group G/Z(G) is not a cyclic group.

We know if G is abelian, Z(G)=G. But now if it is not abelian, can we simply say because G is not cyclic, then any factor group will not be cyclic either? or is there more to it?

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Proof:
If the factor group G/Z(G) is cyclic, then G/Z(G)=<gZ(G)> for some g in G. So for each element in G, it has the ...

Solution Summary

Factor Groups of Non-Abelian Groups are investigated.

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