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Are generators preserved under homomorphism?

  1. Feb 23, 2013 #1
    Sorry if this is such a basic question, but I'm not sure about the answer and having trouble finding it in my text book. If I have a cyclic group G with generator g, and a homomorphism θ: G --> H, does this mean that θ(g) is a generator of H?
  2. jcsd
  3. Feb 23, 2013 #2
    Try proving it yourself, or find a counterexample. To get you started, if [itex]\theta(g) = h[/itex], then what is [itex]\theta(g^2)[/itex]?
  4. Feb 23, 2013 #3
    Is H necessarily cyclic?
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