Editing Group theory (section)

==Connection of groups and symmetry==
{{Main|Symmetry group}}
Given a structured object ''X'' of any sort, a [[symmetry]] is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
*If ''X'' is a set with no additional structure, a symmetry is a [[bijection|bijective]] map from the set to itself, giving rise to permutation groups.
*If the object ''X'' is a set of points in the plane with its [[Metric space|metric]] structure or any other [[metric space]], a symmetry is a [[bijection]] of the set to itself which preserves the distance between each pair of points (an [[isometry]]). The corresponding group is called [[isometry group]] of ''X''.
*If instead [[angle]]s are preserved, one speaks of [[conformal map]]s. Conformal maps give rise to [[Kleinian group]]s, for example.
*Symmetries are not restricted to geometrical objects, but include algebraic objects as well.  For instance, the equation <math>x^2-3=0</math> has the two solutions <math>\sqrt{3}</math> and <math>-\sqrt{3}</math>. In this case, the group that exchanges the two roots is the [[Galois group]] belonging to the equation.  Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.

The axioms of a group formalize the essential aspects of [[symmetry]]. Symmetries form a group: they are [[closure (mathematics)|closed]] because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.

[[Frucht's theorem]] says that every group is the symmetry group of some [[Graph (discrete mathematics)|graph]]. So every abstract group is actually the symmetries of some explicit object.

The saying of "preserving the structure" of an object can be made precise by working in a [[category (mathematics)|category]]. Maps preserving the structure are then the [[morphism]]s, and the symmetry group is the [[automorphism group]] of the object in question.