The work opens by establishing the fundamental conception of an operation applied to a set of objects, defining inverses and the basic criteria that constitute a group. Finite and infinite groups are categorized, with particular attention given to cyclical groups, subgroups, conjugate operations, and various forms of isomorphism such as simple and multiple correspondences. The text then progresses to the theory of continuous groups, distinguishing between finite and infinite types. Developed largely through the work of Sophus Lie, continuous groups are analyzed via infinitesimal operations and their combinants, leading to the classification of simple groups and the introduction of the adjunct group.
Special emphasis is placed on projective groups, contact-transformations, and their extensive applications to differential invariants and the integration of differential equations. Discontinuous groups are explored through generating operations and properly or improperly discontinuous actions, specifically addressing linear discontinuous groups and their significance in the theory of modular functions. Finally, the study turns to groups of finite order, detailing the multiplication table, Sylow's theorem, composition-series, and permutation-groups. The work concludes with the representation of groups as linear substitutions and their critical applications to the theory of algebraic equations, demonstrating how Galois theory utilizes group structures to determine the solvability of equations by radicals.
