Let be a finite subgroup of . For example, is a subgroup of order 4.
(a) Prove that must be a subset of
(b) Prove that is cyclic and is generated by an element of form .
Finite subfields of the multiplicative group of a field must be cyclic subgroups.
From here it should be clear why (a) and (b) follows.
But you can also see (a) because if then it means that for all . But if then if or if . Therefore and so the complex number in must lie on the unit circle. To show (b) let . By Lagrange's theorem we have . Therefore, for some . Define , this subgroup is cyclic and contains , but, , from here it follows that . Thus, must be a cyclic subgroup.