A group homomorphism to the automorphism group of a set X amounts to a group action on X: indeed, each left G-action on a set X determines, and, conversely, each homomorphism defines an action by.
Let be two finite sets of the same cardinality and the set of all bijections. Then, which is a symmetric group, acts on from the left freely and transitively; that is to say, is a torsor for .
Given a field extension, its automorphism group is the group consisting of field automorphisms of L that fixK: it is better known as the Galois group of.
The automorphism group of the projective n-space over a field k is the projective linear group
The automorphism group of a finite-dimensional real Lie algebra has the structure of a Lie group. If G is a Lie group with Lie algebra, then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of.
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.
Let be a finite-dimensional vector space over a fieldk 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 ringR 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.
