Theorem 15.9

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A factor of a cyclic group is cyclic.


Let $G$ be a cyclic generator $a$, and let $N$ be a normal subgroup of $G$. We claim the coset $aN$ generates $G/N$. We must compute all powers of $aN$. But this amounts to computing, in $G$, all powers of the representative $a$ and all these powers give all elements in $G$. Hence the powers of $aN$ certainly give all cosets of $N$ and $G/N$ is cyclic.

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