HINT: Let then is a subgroup of order . Suppose is another subgroup of order . Show .
HINT: For the latter, use the fact that the gcd is a linear combination of and , i.e., Theorem 1.2.3.)
HINT: Write , let , and use Theorem 5.2.2. For uniqueness, show using , that if and satisfy the stated conditions in the problem, then they must be given as stated in this hint.
HINT: Use Theorem 5.2.1.
HINT: Use the Corollary 5.2.3 and Theorem 5.2.4.
(W. Wardlaw) Prove that a finite group is cyclic if and only if has no more than roots of for every , where is the identity of . .
HINT: A root of is a solution to . Use Theorem 5.2.1 for the only if part and the Corollary 5.2.5 for the if part.