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 general linear group of "X", the group of invertible linear transformations from "X" to itself.
Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is called a transformation group (especially in old literature).
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_4 are objects in some category, then the set formula_5 of all formula_6 is a left formula_7-torsor. In practical terms, this says that a different choice of a base point of formula_5 differs unambiguously by an element of formula_7, or that each choice of a base point is precisely a choice of a trivialization of the torsor.
If formula_27 and formula_28 are objects in categories formula_29 and formula_30, and if formula_31 is a functor mapping formula_27 to formula_28, then formula_34 induces a group homomorphism formula_35, 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_36, "C" a category, is called an action or a representation of "G" on the object formula_37, or the objects formula_38. Those objects are then said to be formula_11-objects (as they are acted by formula_11); cf. formula_41-object. If formula_42 is a module category like the category of finite-dimensional vector spaces, then formula_11-objects are also called formula_11-modules.
Automorphism group functor.
Let formula_45 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_46 that preserve the algebraic structure: they form a vector subspace formula_47 of formula_48. The unit group of formula_47 is the automorphism group formula_50. When a basis on "M" is chosen, formula_48 is the space of square matrices and formula_47 is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, formula_50 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_54 preserving the algebraic structure: denote it by formula_55. Then the unit group of the matrix ring formula_55 over "R" is the automorphism group formula_57 and formula_58 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_50.
In general, however, an automorphism group functor may not be represented by a scheme.