Math 3321                                Review Test # 2               Spring 2002

Dr. Cordero

1. Give the following definitions:

·        Transposition

·        Even (odd) permutations

·        Equivalence relation

·        Partition of a set

·        Complete set of equivalence class representatives

·        Greatest common divisor of two integers

·        Relatively prime integers

·        Least common multiple of two integers

·        Standard form of an integer (prime factorization)

·        Subgroup generated by an element

·        Cyclic subgroup

·        Order of an element

1. Describe the subgroup  of .
2. State and prove Theorem 7.2.
3. Describe the subgroups  and of a group of permutations .
4. Show that  and  are subgroups of the group .
5. State in your own words Theorem 9.1 (understand it!).
6. Study the equivalence relation given in Theorem 9.2. (Also, study the proof of Theorem 9.2).
7. State and prove Theorem 10.1.
8. Study the Division Algorithm.
9. Describe the group .
10. What is ?
11. Describe the Euclidean Algorithm.
12. Study Theorem 12.2 and its Corollary.
13. State in your own words the Fundamental Theorem of Arithmetic.
14. State and prove Theorem 14.1.
15. State and prove Theorem 14.2.
16. State Theorem 14.3.