#### 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.