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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, ${\mathrm{SL}}_2({\mathbb{F}}_q)$ consists of determinant-one $2\times 2$ matrices over a finite field. There are $q^2-1$ choices for its first column; for each nonzero first column, the determinant-one equation has $q$ solutions for the second. Dividing by its scalar center gives the projective special linear groupFor $q=5$, its order is sixty. These projective groups are simple for $q>3$, whereas ${\mathrm{PSL}}_2(2)\cong S_3$ and ${\mathrm{PSL}}_2(3)\cong A_4$ are exceptions.^[ref]

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.

James E. Humphreys wrote influential mathematical expositions on Lie theory and representation theory. His book Modular Representations of Finite Groups of Lie Type by James E. Humphreys develops modular representation theory for finite groups of Lie type, relating finite groups to algebraic groups over finite fields. The opening chapter describes constructions using rational points and fixed points of appropriate endomorphisms.^[ref]

A small example illustrates why the characteristic of a field matters in representation theory. Over the finite field ${\mathbb{F}}_2$, the matrixsatisfies $A^2=I$ but $A\ne I$, so it represents the generator of a group of order two. Its fixed vectors form only the line spanned by $(1,0)$. That line has no invariant one-dimensional complement: over ${\mathbb{F}}_2$, an invertible action on a line must be the identity, so two invariant lines would force $A=I$. The example shows concretely how a representation can fail to split into smaller pieces when the field characteristic divides the group order. This is part of the mathematical setting that Humphreys's expositions make accessible.