الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
In this thesis we generalize the concept of differential calculus on hopf algebras by building the calculus on a more generalized structure which is the Hopf group coalgebras. The substance of the thesis is divided into four chapters: Chapter I Presents the main definitions of algebras, coalgebras, the duality between them till we reach the definition of Hopf algebras and Quantum groups. Chapter II Introduces the structure of Hopf group coalgebras , and the module and comodule structure upon it which is the basic tool in our work. Chapter III Presents the basic definition of Differential calculus on Hopf algebras, the concept of left, right, and bicovariant differential calculus. Chapter IV Addresses the structure of first order differential calculus on Hopf group coalgebras, the definitions of left right and bicovariant pi graded bimodules