Are generators preserved under homomorphism?

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In the discussion about whether generators are preserved under homomorphism, the main question revolves around a cyclic group G with generator g and a homomorphism θ: G --> H. It is questioned whether θ(g) serves as a generator for H. Participants are encouraged to either prove this assertion or provide a counterexample, prompting exploration of the implications of θ(g) = h and the behavior of θ(g^2). Additionally, the cyclic nature of group H is questioned, leading to further examination of the properties of homomorphisms. The conversation highlights the complexity of group theory and the nuances of how generators behave under homomorphic mappings.
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Sorry if this is such a basic question, but I'm not sure about the answer and having trouble finding it in my textbook. 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?
 
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Try proving it yourself, or find a counterexample. To get you started, if \theta(g) = h, then what is \theta(g^2)?
 
Is H necessarily cyclic?
 
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