Are generators preserved under homomorphism?

[dumbQuestion]

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?

 

[Number Nine]

Try proving it yourself, or find a counterexample. To get you started, if $\theta(g) = h$, then what is $\theta(g^2)$?

 

[micromass]

Is H necessarily cyclic?