On top of these 18 families, there are 26 individual groups called sporadic groups, with the largest being the Monster weighing in at over 800 trillion elements!
After the classification was announced, some people thought that finite group theory had reached its end. However, many mathematicians continue to work in this area today by simplifying and consolidating parts of the proof, which is currently being done in a series of books published by the American Mathematical Society. The extension problem for finite groups is another major theme, where chemists want to understand how atoms can combine to form bigger molecules. Infinite groups come in many forms, but some mathematicians are looking for tame subclasses that might be amenable to analysis using techniques from the classification of finite simple groups.

The original answer provided a brief overview of the classification of finite simple groups and their infinite number. However, it did not explain how this classification is useful in mathematics or provide any context on its significance beyond being analogous to regular polyhedra’s rotation groups.

To refine the answer for better understanding, let us dig deeper into the topic. The classification of finite simple groups provides a complete list of all the finite simple groups, which are abstract analogues of the rotation groups of regular polyhedra. Despite their infinite number, mathematicians understand them well and have precise descriptions for 18 infinite families of finite simple groups obtained from automorphism groups of certain geometric structures. On top of these 18 families, there are 26 individual groups called sporadic groups, with the largest being the Monster weighing in at over 800 trillion elements!

This classification is significant because it provides a framework for understanding and analyzing finite simple groups, which have many applications in various fields of mathematics. For example, they are used to study algebraic structures such as rings, modules, and vector spaces; to solve problems in number theory, combinatorics, and topology; and to develop new techniques for solving complex mathematical equations.

After the classification was announced, some people thought that finite group theory had reached its end. However, many mathematicians continue to work in this area today by simplifying and consolidating parts of the proof, which is currently being done in a series of books published by the American Mathematical Society. The extension problem for finite groups is another major theme, where chemists want to understand how atoms can combine to form bigger molecules. Infinite groups come in many forms, but some mathematicians are looking for tame subclasses that might be amenable to analysis using techniques from the classification of finite simple groups.