More generally, any group of Lie type has the structure of a BN-pair.

BN-pairs can be used to prove that most groups of Lie type are simple.

These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

G(4) is an exceptional group of Lie type.

Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way.

Alternating groups sometimes behave as if they were groups of Lie type over the field with one element.

There are coincidences between alternating groups and small groups of Lie type:

Similarly, many groups of Lie type are Hurwitz.

Their finite analogues are the classical groups of Lie type.

When the underlying ring is a finite field the classical groups are groups of Lie type.