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I am stuck on trying to find an element in $G/H = \{g + H | g \in G\}$ that will generate the whole set. My intuition tells me it is true since $G$ and $H$ are cyclic. I have a feeling that this direct approach of definition of a cyclic group is not the effective way to tackle this problem. Any advice appreciated, thank you.

Lutterbach
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    If $G=\langle g\rangle$ then $gH$ generates $G/H$ – David P Dec 07 '19 at 06:58
  • This question has been asked many times and is easy to find by googling. You should try that before asking questions. – verret Dec 07 '19 at 22:23
  • @verret I did in fact google it and must have missed your link (which was the only link from stack exchange related to this problem, asked 4 years ago). Thank you for the condescending remark. – Lutterbach Dec 07 '19 at 23:15
  • You may consider my remark condescending, but due diligence before asking a question is actually site policy. – verret Dec 08 '19 at 00:17
  • Yes I understand; however, there are ways to inform others without being condescending. – Lutterbach Dec 08 '19 at 03:50

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If $g$ is a generator for $G$, $gH$ (the coset) by definition generates $G{/}H$, as $x \to xH$ is a surjective homomorphism.

Henno Brandsma
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