## Exercises

1. If and are finite cyclic groups such that and is defined as in the text. Show is 1-1, onto, and operation preserving.

2. Let with . Prove that has a unique subgroup of order where .

HINT: Let then is a subgroup of order . Suppose is another subgroup of order . Show .

3. Let with . Let be integers such that (). Suppose and . Prove that and .

HINT: For the latter, use the fact that the gcd is a linear combination of and , i.e., Theorem 1.2.3.)

4. Let be a group and such that where . Show that one can write where , , and . Moreover, prove the uniqueness of such a representation.

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.

5. Prove that a group of order , where is a prime and , must contain a subgroup of order .

HINT: Use Theorem 5.2.1.

6. If in a group of order , for each positive , the equation has less than solutions, then show is cyclic.

HINT: Use the Corollary 5.2.3 and Theorem 5.2.4.

7. (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.

David Joyner 2007-08-06