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How to define a subgroup and get its generators?

I am trying implement the Asychronous Group Auhtnetication Scheme with Multiple Authentication,and i was stucked. Suppose that We get two big primes p and q and q divides p-1. GF(q) is a unique subgroup of GF(p) with order q,and every gi is a generator of GF(p). i do not know how to construct a subgroup GF(q) meets the condition above .What makes me more confused that i remember GF(p) only has one generator 1, if i was wrong, how to get those generators gi?image description