Find all the finite subgroups of Q set of all nonzero ratio

Find all the finite subgroups of Q * (set of all nonzero rational numbers) under multiplication. For each of these subgroups, answer the question: is the subgroup cyclic? If yest, name all its generators; if no, explain why. Explain how you know that Q * has no other finite subgroups

Solution

H={1} and K={-1, 1} are only two finite subgroups of Q* under multiplication. Both are cyclic as both H and K have orders 1 and 2. In H generaor is 1 and in K it is -1. it is quite obvious from its stake up property. i.e. <2>={2m, m is in Q*}.