Easy question in algebra

Hi,

I suppose this is an easy question in algebra:
Let p,q be prime. Let further p=2q+1. Let g be an element of order q of the multiplicative group Z_p^*. So g is a generator of the order-q subgroup of Z_p^*. Every element h=g^x, for x \in {1, ..., q-1} is a generator of G. Is this correct? I suppose yes. Why is it not correct for x= q? Is the reason for that, that g^q =1 and 1 is not a generator of G?

Thanks for your help
.

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