Are generators preserved under homomorphism?

1. Feb 23, 2013

dumbQuestion

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. Feb 23, 2013

Number Nine

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

3. Feb 23, 2013

micromass

Is H necessarily cyclic?