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New to group theory or abstract algebra, this is the right book for you. This group has a lot of basic definitions of group theory terms with examples and a step by step prove of basic theorems/lemmas in the preliminaries. It has a chapter on nilpotent groups, which gives definition of a nilpotent group,examples of a nilpotent groups and some step by step proves of lemmas/theorem. It also has a chapter on solvable groups which gives definition of solvable group,examples of a solvable groups and some step by step proves of lemmas/theorem. The Hall and Schur-Zassenhaus Theorems defined and examples given with proves of lemmas/theorems. Carter subgroup defined and examples of groups with cater subgroups given with explanations. Examples of groups with no Cater subgroup was also given with detailed explanation. A detailed and step by step prove of Carter's theorem given.