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  "body": "\n## Definition\n\nLet $p$ be a prime and let $k$ be an [[algebraic closure|0X6AB5P3G1]] of the [[finite field|0ZXZ82FTX2]] $\\F_p$.\nLet $\\mathbf{G}$ be a connected reductive linear algebraic group over $k$.\nA Steinberg endomorphism of $\\mathbf{G}$ is an endomorphism $F$ of the algebraic group $\\mathbf{G}$ with the following property.\nSome power $F^m$ with $m \\ge 1$ is the [[Frobenius endomorphism|0T3WVTD675]] of $\\mathbf{G}$ for some structure of $\\mathbf{G}$ over a finite subfield of $k$.\nA *finite group of Lie type* is a group\n$$\\mathbf{G}^F = \\set{g \\in \\mathbf{G} : F(g) = g}$$\nfor such $\\mathbf{G}$ and $F$.\n\n## Remarks\n\nThe group $\\mathbf{G}^F$ is always finite.\nExample: take $\\mathbf{G} = \\mathrm{GL}_n(k)$ and let $F$ raise every matrix entry to the power $q$, where $q$ is a power of $p$.\nThen $F$ is the Frobenius endomorphism for the standard structure over $\\F_q$, and $\\mathbf{G}^F = \\mathrm{GL}_n(\\F_q)$, since the elements of $k$ fixed by $x \\mapsto x^q$ are exactly the elements of $\\F_q$.\nNon-example: $\\mathrm{GL}_n(\\R)$ for $n \\ge 1$ is not a finite group of Lie type, since it is infinite.\nMany authors also call the simple groups obtained from $\\mathbf{G}^F$, such as $\\mathrm{PSL}_n(\\F_q)$, groups of Lie type.\nThe [[Chevalley groups|YPCH4Q0V34]] over $\\F_q$ come from the case where $\\mathbf{G}$ is split over $\\F_q$ and $F$ is the corresponding Frobenius endomorphism.\nThe twisted groups, such as the unitary groups, come from the other Steinberg endomorphisms.\nFor the [[Suzuki groups|36X3GET13B]] and the [[Ree groups|YB62XEQJAC]], the map $F$ is not a Frobenius endomorphism, but $F^2$ is one.\n\n## Explanations\n\n### L1 human\n\nA finite group of Lie type is a matrix group over a finite field, or a twisted variant of one, described as the fixed points of a Frobenius-type map.\n",
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