This subtopic studies foundations of group theory, including basic definitions, constructions, and the abstract principles underlying group-based algebra.

This subtopic studies permutation groups, focusing on group actions, transitivity, primitive groups, and symmetry in combinatorial settings.

This subtopic studies representation theory of groups, where groups are analyzed through linear actions on vector spaces and modules.

This subtopic studies abstract finite groups, including classification themes, solvability, and the structural properties of finite group systems.

This subtopic studies structure and classification of groups, especially decompositions, composition series, and structural invariants that organize group theory.

This subtopic studies infinite groups and combinatorial group theory, emphasizing presentations, generators and relations, and geometric or algorithmic methods.

This subtopic studies linear algebraic groups, where group structure is studied through algebraic varieties and matrix representations.

This subtopic studies matrix groups, including classical matrix groups and their algebraic and geometric properties.

This subtopic studies connections with homological algebra, where group-theoretic questions are analyzed using homology, cohomology, and derived constructions.

This subtopic studies abelian groups, focusing on classification, torsion phenomena, and module-theoretic viewpoints.

This subtopic studies semigroups, examining associative binary operations without inverses and the algebraic structure that arises from them.

This subtopic studies other generalizations of groups, including quasigroups, loops, and related systems that weaken group axioms.

This subtopic studies probabilistic methods in group theory, using random processes and probabilistic arguments to study group structure and behavior.