Solvable and Nilpotent Groups

Abstract

This thesis introduces the concepts of solvable and nilpotent groups and presents important definitions like supersolvable groups, polycyclic groups, Carter subgroups, metabelian groups ,hypercentral groups, HM= groups, and Chernicov groups. This thesis includes applications in Galois theory, and the solvability by radicals. Also, this research presents examples, propositions, and applications related to solvable and nilpotent groups. Finally, this thesis presents three applications of solvable and nilpotent groups, it discusses the solvability of Carter subgroups in the groups in which every element is conjugate to its inverse, proofs of interesting properties of metabelian groups, and this thesis study groups with many hypercentral subgroups.