The Exploring the Category of Groups: Functorial Methods and Applications

Abstract

In this study the category-theoretic perspective on groups is investigated, with a particular focus on functorial procedures as a unifying approach to the comprehension of algebraic structures.  Beginning with the construction of the category of groups, we investigate functors between group-theoretic categories, adjoint functors, and universal qualities all the way through to the end of the discussion.  For the purpose of demonstrating how functorial approaches enrich the study of group theory and provide bridges between abstract algebra and other mathematical areas, applications are examined in the fields of representation theory, group cohomology, and topology.