The present book is an English translation from my book with the same title in the Arabic language which is based on my lectures given to students of various colleges studying mathematics.This book consists of seven chapters. Chapter 1, "Sets - Relations -Functions" in abstract algebra, Chapter 2 contains the "Groups" as one of the main subjects. In Chapter 3, we will discuss "Permutation Group" as a practical part and very useful in Linear Algebra. Chapter 4 presents "Isomorphism" which a fundamental part, and has many applications. Chapter 5, contains "The Natural Numbers" and how to extend the natural numbers up to the real filed. Chapter 6 contains " Rings - Fields" as the second main subject to abstract algebra. Chapter 7, which deal with "Continuation on Groups".