Automorphism group
In mathematics, the automorphism group of an object "X" is the group consisting of automorphisms of "X". For example, if "X" is a finite-dimensional vector space, then the automorphism group of "X" is the group of invertible linear transformations from "X" to itself (the general linear group of "X"). If instead "X" is a group, then its automorphism group formula_1 is the group consisting of all group automorphisms of "X". Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups  are studied in a general way in the field of category theory. Examples. If "X" is a set with no additional structure, then any bijection from "X" to itself is an automorphism, and hence the automorphism group of "X" in this case is precisely the symmetric group of "X". If the set "X" has additional structure, then it may be the case that not all bijections on the set preserve this structure, in which case the automorphism group will be a subgroup of the symmetric group on "X". Some examples of this include the following: If "G" is a group  acting on a set "X", the action amounts to a group homomorphism from "G" to the automorphism group of "X" amounts to a group action on "X". Indeed, each left "G"-action on a set "X" determines formula_11, and, conversely, each homomorphism formula_12 defines an action by formula_13. This extends to the case when the set "X" has more structure than just a set. For example, if "X" is a vector space, then a group action of "G" on "X" is a "group representation" of the group "G", representing "G" as a group of linear transformations (automorphisms) of "X"; these representations  are the main object of study in the field of representation theory. Here are some other facts about automorphism groups: In category theory. Automorphism groups appear very naturally in category theory. If "X" is an object in a category, then the automorphism group of "X" is the group consisting of all the invertible morphisms from "X" to itself. It is the unit group of the endomorphism monoid of "X". (For some examples, see PROP.) If formula_14 are objects in some category, then the set formula_15 of all formula_16 is a left formula_17-torsor. In practical terms, this says that a  different choice of a base point of formula_15 differs unambiguously by an element of formula_17, or that each choice of a base point is precisely a choice of a trivialization of the torsor. If formula_28 and formula_29 are objects in categories formula_30 and formula_31, and if formula_32 is a functor mapping formula_28 to formula_29, then formula_35 induces a group homomorphism formula_36, as it maps invertible morphisms to invertible morphisms. In particular, if "G" is a group viewed as a category with a single object * or, more generally, if "G" is a groupoid, then each functor formula_37, "C" a category, is  called an action or a representation of "G" on the object formula_38, or the objects formula_39. Those objects are then said to be formula_4-objects (as they are acted by formula_4); cf. formula_42-object. If formula_43 is a module category like the category of finite-dimensional vector spaces, then formula_4-objects are also called formula_4-modules. Automorphism group functor. Let formula_46 be a finite-dimensional vector space over a field "k" that is equipped with some algebraic structure (that is, "M" is a finite-dimensional algebra over "k"). It can be, for example, an associative algebra or a Lie algebra . Now, consider "k"-linear maps formula_47 that preserve the algebraic structure: they form a vector subspace formula_48 of formula_49. The unit group of formula_48 is the automorphism group formula_51. When a basis on "M" is chosen, formula_49 is the space of square matrices and formula_48 is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, formula_51 is a linear algebraic group over "k". Now base extensions applied to the above discussion determines a functor: namely, for each commutative ring "R" over "k", consider the "R"-linear maps formula_55 preserving the algebraic structure: denote it by  formula_56. Then the unit group of the matrix ring formula_56 over "R" is the automorphism group formula_58 and formula_59 is a group functor: a functor from the category of commutative rings over "k" to the category of groups. Even better, it is represented by a scheme (since the automorphism groups are defined by polynomials): this scheme is called the automorphism group scheme and is denoted by formula_51. In general, however, an automorphism group functor may not be represented by a scheme.