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 (the general linear group of X). If instead X is a group, then its automorphism group $$\operatorname{Aut}(X)$$ 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:
 * The automorphism group of a field extension $$L/K$$ is the group consisting of field automorphisms of L that fix K. If the field extension is Galois, the automorphism group is called the Galois group of the field extension.
 * The automorphism group of the projective n-space over a field k is the projective linear group $$\operatorname{PGL}_n(k).$$
 * The automorphism group $$G$$ of a finite cyclic group of order n is isomorphic to $$(\mathbb{Z}/n\mathbb{Z})^\times$$, the multiplicative group of integers modulo n, with the isomorphism given by $$\overline{a} \mapsto \sigma_a \in G, \, \sigma_a(x) = x^a$$. In particular, $$G$$ is an abelian group.
 * The automorphism group of a finite-dimensional real Lie algebra $$\mathfrak{g}$$ has the structure of a (real) Lie group (in fact, it is even a linear algebraic group: see below). If G is a Lie group with Lie algebra $$\mathfrak{g}$$, then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of $$\mathfrak{g}$$.

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 $$G \to \operatorname{Aut}(X), \, g \mapsto \sigma_g, \, \sigma_g(x) = g \cdot x$$, and, conversely, each homomorphism $$\varphi: G \to \operatorname{Aut}(X)$$ defines an action by $$g \cdot x = \varphi(g)x$$. 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:
 * Let $$A, B$$ be two finite sets of the same cardinality and $$\operatorname{Iso}(A, B)$$ the set of all bijections $$A \mathrel{\overset{\sim}\to} B$$. Then $$\operatorname{Aut}(B)$$, which is a symmetric group (see above), acts on $$\operatorname{Iso}(A, B)$$ from the left freely and transitively; that is to say, $$\operatorname{Iso}(A, B)$$ is a torsor for $$\operatorname{Aut}(B)$$ (cf. ).
 * Let P be a finitely generated projective module over a ring R. Then there is an embedding $$\operatorname{Aut}(P) \hookrightarrow \operatorname{GL}_n(R)$$, unique up to inner automorphisms.

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 $$A, B$$ are objects in some category, then the set $$\operatorname{Iso}(A, B)$$ of all $$A \mathrel{\overset{\sim}\to} B$$ is a left $$\operatorname{Aut}(B)$$-torsor. In practical terms, this says that a different choice of a base point of $$\operatorname{Iso}(A, B)$$ differs unambiguously by an element of $$\operatorname{Aut}(B)$$, or that each choice of a base point is precisely a choice of a trivialization of the torsor.

If $$X_1$$ and $$X_2$$ are objects in categories $$C_1$$ and $$C_2$$, and if $$F: C_1 \to C_2$$ is a functor mapping $$X_1$$ to $$X_2$$, then $$F$$ induces a group homomorphism $$\operatorname{Aut}(X_1) \to \operatorname{Aut}(X_2)$$, 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 $$F: G \to C$$, C a category, is called an action or a representation of G on the object $$F(*)$$, or the objects $$F(\operatorname{Obj}(G))$$. Those objects are then said to be $$G$$-objects (as they are acted by $$G$$); cf. $\mathbb{S}$-object. If $$C$$ is a module category like the category of finite-dimensional vector spaces, then $$G$$-objects are also called $$G$$-modules.

Automorphism group functor
Let $$M$$ 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 $$M \to M$$ that preserve the algebraic structure: they form a vector subspace $$\operatorname{End}_{\text{alg}}(M)$$ of $$\operatorname{End}(M)$$. The unit group of $$\operatorname{End}_{\text{alg}}(M)$$ is the automorphism group $$\operatorname{Aut}(M)$$. When a basis on M is chosen, $$\operatorname{End}(M)$$ is the space of square matrices and $$\operatorname{End}_{\text{alg}}(M)$$ is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, $$\operatorname{Aut}(M)$$ 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 $$M \otimes R \to M \otimes R$$ preserving the algebraic structure: denote it by $$\operatorname{End}_{\text{alg}}(M \otimes R)$$. Then the unit group of the matrix ring $$\operatorname{End}_{\text{alg}}(M \otimes R)$$ over R is the automorphism group $$\operatorname{Aut}(M \otimes R)$$ and $$R \mapsto \operatorname{Aut}(M \otimes R)$$ 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 $$\operatorname{Aut}(M)$$.

In general, however, an automorphism group functor may not be represented by a scheme.