The Enormous Theorem

In the early 1960s, Group Theory was a way of describing several phenomena in a concise manner.

Morton Hamermesh's book is great.  Chapter 1 is "Elements of Group Theory".

The applications of Group Theory have moved to "Representation Theory".  The theory side seems to have evolved to Category Theory.

Category theory can be used to formalize concepts of other high-level abstractions such as set theory, ring theory, and group theory. Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself. In category theory, a "morphism" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made.

Morphism In mathematical category theory, a generalization or abstraction of the concept of a structure-preserving function.

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

In category theory, morphism is a broadly similar idea, but somewhat more abstract:  the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The July 2015 issue of Scientific American on page 72 summarizes "Four Enormous Families" that contain all the finite simple groups.  An exemplar for completeness. 

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