Editing Symmetry group (section)

==Symmetry groups in general==
{{see also|Automorphism|Automorphism group}}
In wider contexts, a '''symmetry group''' may be any kind of '''transformation group''', or [[automorphism]] group. Each type of [[mathematical structure]] has [[Bijection|invertible mappings]] which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the [[Erlangen programme]].

For example, objects in a hyperbolic [[non-Euclidean geometry]] have [[Fuchsian group|Fuchsian symmetry groups]], which are the discrete subgroups of the isometry group of the hyperbolic plane, preserving hyperbolic rather than Euclidean distance. (Some are depicted in drawings of [[M. C. Escher|Escher]].) Similarly, automorphism groups of [[finite geometry|finite geometries]] preserve families of point-sets (discrete subspaces) rather than Euclidean subspaces, distances, or inner products. Just as for Euclidean figures, objects in any geometric space have symmetry groups which are subgroups of the symmetries of the ambient space.

Another example of a symmetry group is that of a [[Graph (discrete mathematics)|combinatorial graph]]: a graph symmetry is a permutation of the vertices which takes edges to edges. Any [[Presentation of a group|finitely presented group]] is the symmetry group of its [[Cayley graph]]; the [[free group]] is the symmetry group of an infinite [[Tree (graph theory)|tree graph]].