This book covers a wide range of topics in abstract algebra. It explores the properties and characteristics of finite quasi-core-p p-groups, shedding light on their unique structures. Additionally, the book explores into the computer-aided analysis of solvable rigid Lie algebras with a given eigenvalue spectrum, offering insights into their solvability and rigidity.The book concludes with a chapter on the left (right) regular elements of some transformation semigroups, providing a deeper understanding of their properties and applications. Finally, the book explores the parity properties of configurations, offering insights into the characteristics and symmetries of these mathematical arrangements. This book is intended for readers with a background in mathematics, particularly those interested in abstract algebra and its applications. It caters to undergraduate and graduate students studying mathematics, as well as researchers and professionals working in fields where abstract algebra is applied, such as cryptography, coding theory, and theoretical computer science. The book assumes a basic understanding of algebraic structures and mathematical proofs, making it suitable for advanced study and research in abstract algebra.