If G is Cyclic so is G/H Proof

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If G is Cyclic so is G/H Proof. In this video we prove that if G is a cyclic group then the quotient group G/H is also cyclic. It turns out that if g is a generator for G, then Hg is a generator for G/H.
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How do I prove that the quotient group itself is a group?

goodluckendoscopy
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give me example of cyclic quotient group

sazankochar