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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.