A finite group of Lie type is built from an algebraic group in positive characteristic by taking fixed points of a suitable Frobenius-type endomorphism, often followed by a derived-group or central-quotient construction. The terminology covers classical matrix families and exceptional families, including twisted forms. It connects the structure of Lie theory with finite group theory; such a finite group is not a positive-dimensional real Lie group.[ref]
For example, consists of determinant-one matrices over a finite field. There are choices for its first column; for each nonzero first column, the determinant-one equation has solutions for the second. Dividing by its scalar center gives the projective special linear groupFor , its order is sixty. These projective groups are simple for , whereas and are exceptions.[ref]
Simplicity of PSLn(F) by Keith Conrad. Introduction and section 3
The classification of finite simple groups uses the simple members of the Lie-type families, with exceptional small parameters treated explicitly. A family name alone does not guarantee simplicity.
Articles by others on the same topic
Groups of Lie type are a class of algebraic groups that can be associated with simple Lie algebras and are defined over finite fields. They play a significant role in the theory of finite groups, particularly in the classification of finite simple groups. The concept of groups of Lie type arises from the representation theory of Lie algebras over fields, especially over finite fields.
In the classification of finite simple groups, groups of Lie type are a set of infinite families of simple lie groups. These are the other infinite families besides te cyclic groups and alternating groups.
A decent list at: en.wikipedia.org/wiki/List_of_finite_simple_groups, en.wikipedia.org/wiki/Group_of_Lie_type is just too unclear. The groups of Lie type can be subdivided into:
- Chevalley groups
- TODO the rest
The first in this family discovered were a subset of the Chevalley groups by Galois: , so it might be a good first one to try and understand what it looks like.
TODO understand intuitively why they are called of Lie type. Their names , seem to correspond to the members of the classification of simple Lie groups which are also named like that.
But they are of course related to Lie groups, and as suggested at Video "Yang-Mills 1 by David Metzler (2011)" part 2, the continuity actually simplifies things.