Editing Group theory (section)

===Lie theory===
{{Main|Lie theory}}
A [[Lie group]] is a [[group (mathematics)|group]] that is also a [[differentiable manifold]], with the property that the group operations are compatible with the [[Differential structure|smooth structure]]. Lie groups are named after [[Sophus Lie]], who laid the foundations of the theory of continuous [[transformation group]]s. The term ''groupes de Lie'' first appeared in French in 1893 in the thesis of Lie's student [[:pt:Arthur Tresse|Arthur Tresse]], page 3.<ref>{{citation |title= Sur les invariants différentiels des groupes continus de transformations | author= Arthur Tresse |journal=Acta Mathematica|volume=18|year=1893|pages=1–88 |doi=10.1007/bf02418270|url=https://zenodo.org/record/2273334|doi-access=free}}</ref>

Lie groups represent the best-developed theory of [[continuous symmetry]] of [[mathematical object]]s and [[mathematical structure|structures]], which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern [[theoretical physics]]. They provide a natural framework for analysing the continuous symmetries of [[differential equations]] ([[differential Galois theory]]), in much the same way as permutation groups are used in [[Galois theory]] for analysing the discrete symmetries of [[algebraic equations]]. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.