"G is a cyclic group with normal subgroup N. Show that G/N is cyclic."
Not sure what to do here. I know that all subgroups of a cyclic group are also cyclic. But the quotient group is confusing me here.
First of all, note that cyclic groups are of necessity abelian. So ANY subgroup is automatically normal.
Since G is cyclic, there is some element g of G that generates G. I claim gN generates G/N.
For suppose xN is ANY element of G/N. Since x is some element of G, we have , for some integer k (since G is cyclic, with generator g).
Thus: , which shows that G/N is generated by gN.