On the construction and classification of ‘small’ groups

In this talk, we will review how each finite group can be disassembled into a unique set of "simple" groups, similar to the unique prime factorization of natural numbers. This raises various questions; for example: which simple pieces are there? And: in which ways can one glue these pieces together to form new finite groups? The main focus of this talk will be on this last question, and we will discuss it in the context of classifying (almost) all groups of order at most 20,000.