Editing Group theory (section)

===Transformation groups===
Permutation groups and matrix groups are special cases of [[transformation group]]s: groups that act on a certain space ''X'' preserving its inherent structure. In the case of permutation groups, ''X'' is a set; for matrix groups, ''X'' is a [[vector space]]. The concept of a transformation group is closely related with the concept of a [[symmetry group]]: transformation groups frequently consist of ''all'' transformations that preserve a certain structure.
  
The theory of transformation groups forms a bridge connecting group theory with [[differential geometry]]. A long line of research, originating with [[Sophus Lie|Lie]] and [[Felix Klein|Klein]], considers group actions on [[manifold]]s by [[homeomorphism]]s or [[diffeomorphism]]s. The groups themselves may be [[discrete group|discrete]] or [[continuous group|continuous]].