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# MATH 300

BS Mathematics

5

6

30

30

0

## Content

#### Objective

The aim of this course is to provide the student the basic knowledge about the theory of groups. . Pedagogical Objectives : At the end of this course, the student should be able to: 1. Recognize a structure of group on a set and give examples of groups. 2. Prove the elementary properties of groups. 3. Build the table of some finite groups. 4. Prove that a non-empty subset is a subgroup. 5. Prove that an application of a group in another is a homomorphism or an isomorphism of groups. 6. Build some Isomorphism of groups. 7. Use the isomorphism theorems. 8. Determine the kernel of a homomorphism of groups. 9. Prove that a subgroup is normal and construct the quotient group. 10. Determine the group product of two groups. 11. Determine the order of an element in a finite group. 12. Use the properties of cyclic groups. 13. Determine the parity of a permutation and the decomposition in product of disjoint cycles. 14. Use the main results concerning the action of a group on a set. 15. Use the main results concerning the p-groups. 16. Use the main results concerning the Sylow’s theorems. 17. Use the main results concerning solvable groups. 18. Know that the group Sn is solvable for n  {1,2,3,4} only.

#### Content

Examples of groups : (ℤ , +), (ℤ/nℤ , +), orthogonal group, group of nth roots of the unit, dihedral group, group of quaternions and other groups. . Groups : subgroup generated by a subset, order of an element, properties and classifications of the cyclic groups, finite group, quotient and isomorphism theorems, direct product of groups and internal direct product of subgroups. Permutation group Sn : permutation, transposition, cycle, signature and parity of a permutation, alternating group An , decomposition of a permutation as a product of disjoint cycles, normal subgroup of Sn, simplicity of An for n > 5. . Action on groups : Definition and examples, orbit and stabilizer and its properties, conjugation, the class equation. . Sylow theorems : structure of p-groups, Sylow’s theorems, classification of finite abelian groups. Solvable group : Definition of the derivative of a subgroup of a group and definition of solvable groups, examples and properties, Sn not solvable for n > 5.

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