In this semester I'm taking a lecture about Lie algebra. I'm quite surprised because theory of Lie algebra has very similar structure as theory of groups. Here is some of my observations:
Notion of nilpotent and solvable is almost same and their property is similar.
Theory of representation is also similar. For example, Schur's Lemma holds for representation of groups and Lie algebras.
Classification of Lie algebras and Finite groups has similar parts, like exceptional groups of Lie type and some matrix groups.
They are just some observations so it can be meaningless set of phenomenons. But, do you know any systematic reason of this phenomenon, then please tell me something. I have just naive explanation: commutator of groups and Lie brakets has very similar properties. Maybe Lie algebra - Lie group correspondence has something to do with, but in my intuition finite dimensional Lie algebra is similar with finite groups and Lie groups are infinite, so it may not be ultimate answer. Is there any systematic reason?