Automorphism group


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. If instead X is a group, then its automorphism group 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 and conversely. Indeed, each left G-action on a set X determines, and, conversely, each homomorphism defines an action by. 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 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.
If are objects in some category, then the set of all is a left -torsor. 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.
If and are objects in categories and, and if is a functor mapping to, then induces a group homomorphism, 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, C a category, is called an action or a representation of G on the object, or the objects. Those objects are then said to be -objects ; cf. -object. If is a module category like the category of finite-dimensional vector spaces, then -objects are also called -modules.

Automorphism group functor

Let be a finite-dimensional vector space over a field k that is equipped with some algebraic structure. It can be, for example, an associative algebra or a Lie algebra.
Now, consider k-linear maps that preserve the algebraic structure: they form a vector subspace of. The unit group of is the automorphism group. When a basis on M is chosen, is the space of square matrices and is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, 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 preserving the algebraic structure: denote it by. Then the unit group of the matrix ring over R is the automorphism group and 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 : 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.