Besides the understandable Wikipedia definition, Video 171. ""Simple Groups - Abstract Algebra" by Socratica (2018)" gives an understandable one:
Given a finite group and a simple group , find all groups such that is a normal subgroup of and .
We don't really know how to make up larger groups from smaller simple groups, which would complete the classification of finite groups:
In particular, this is hard because you can't just take the direct product of groups to retrieve the original group: Section "Relationship between the quotient group and direct products".
- Classification of finite groups | 69, 1k, 27
- Relationship between the quotient group and direct products | 308
- Semidirect product | 441