1.Reviewing of some standard basic topics on group product ( direct and semidirect product) and on commutator subgroup and its properties. Isomorphism Theorems 2. Group actions Orbits, stabilizers. Sylow's Theorems and applications. Burnside's formula and permutation character. Induced action. Simplicity of some groups 3. Generators and Relations. Generators, Frattini subgroup (Schur's theorem).Finitely generated abelian groups. Free groups. Word problem. 4. Nilpotent and soluble groups. Central series and nilpotent groups, Fitting subgroup. p-nilpotent groups. Fixed point free automorphims and Frobenius Groups. Soluble groups; Carter subgroups. Schmidt-Iwasawa's Theorem.