Group of Lie type
= Group of Lie type
{title2=$G^F=\{g:F(g)=g\}$}
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>.<Modular Representations of Finite Groups of Lie Type by James E. Humphreys>[Chapter 1, sections 1.1 and 1.3]{ref}
For example, $\operatorname{SL}_2(\mathbb F_q)$ consists of determinant-one $2\times2$ 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 group>
$$
|\operatorname{PSL}_2(q)|=\frac{q(q^2-1)}{\gcd(2,q-1)}.
$$
For $q=5$, its order is sixty. These projective groups are simple for $q>3$, whereas $\operatorname{PSL}_2(2)\cong S_3$ and $\operatorname{PSL}_2(3)\cong A_4$ are exceptions.<Simplicity of PSLn(F) by Keith Conrad>[Introduction and section 3]{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.