Automorphism group

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}.{{efn|First, if G is simply connected, the automorphism group of G is that of \mathfrak{g}. Second, every connected Lie group is of the form \widetilde{G}/C where \widetilde{G} is a simply connected Lie group and C is a central subgroup and the automorphism group of G is the automorphism group of G that preserves C. Third, by convention, a Lie group is second countable and has at most coutably many connected components; thus, the general case reduces to the connected case.}} 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. #In category theory). • 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. == See also ==