In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms.
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).
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.
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 and conversely.
Indeed, each left G-action on a set X determines
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: 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.
are objects in some category, then the set
In practical terms, this says that a different choice of a base point of
differs unambiguously by an element of
, or that each choice of a base point is precisely a choice of a trivialization of the torsor.
induces a group homomorphism
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
, C a category, is called an action or a representation of G on the object
is a module category like the category of finite-dimensional vector spaces, then
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).
that preserve the algebraic structure: they form a vector subspace
is the space of square matrices and
is a linear algebraic group over k. Now base extensions applied to the above discussion determines a functor:[6] namely, for each commutative ring R over k, consider the R-linear maps
preserving the algebraic structure: denote it by
Then the unit group of the matrix ring
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
In general, however, an automorphism group functor may not be represented by a scheme.