In a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.

Group theory is applied in Physics, Chemistry, Crystallography and many further fields.
Discrete and continuous groups are relevant for probability theory, graph theory, in the theory of games and as well in economical and statistical sciences.

Discrete, Finite and Lie Groups

Comprehensive Group Theory in Geometry and Analysis

Pietro Giuseppe Fré

In the series De Gruyter STEM

This is the most recent of my scientific groups and it aims at an original presentation of Group Theory which I thought many times in my life, at Torino University in the years 1987-1989, at SISSA in Trieste in the years 1990-1996 and once again at Torino University in the last three years of my academic career 2020-2022. Furthermore Group Theory in its various aspects has been a central item in all my different Research Projects.
The originality resides in that on one hand Groups are discussed from stretch in all of their aspects and capicities (Discrete, Finite and Lie versions) and are seen as a transversal fundamental ingredient of various branches of Mathematics (Analysis, Geometry, Topology, Combinatorics). Moreover they are essential not only to Physics but also to other provinces of human knowledge.
An important complement to the book is the rich  library of Wolfram MATHEMATICA NoteBooks devoted to Group Theoretical Constructions that can be downloaded from the De Gruyter site whose working and goals are described in appropriate appendices of the book itself.
The new book Discrete, Finite and Lie groups has a partial overlap with Groups and Manifolds but it is different in conceptiont, pursued goals and target readers. Indeed while Group and Manifolds is a TextBook for Physics - major graduate students, the new book is conceived as an advanced reading for scholars and students of various disciplines and it aims at emphasizing the transversal role Group Theory in all branches of Mathematics including analysis. The library of Mathematica NoteBooks has been completely revised updated and systematized.