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 Aut --> ( X ) {\displaystyle \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.
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 G → → --> Aut --> ( X ) , g ↦ ↦ --> σ σ --> g , σ σ --> g ( x ) = g ⋅ ⋅ --> x {\displaystyle G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x} , and, conversely, each homomorphism φ φ --> : G → → --> Aut --> ( X ) {\displaystyle \varphi :G\to \operatorname {Aut} (X)} defines an action by g ⋅ ⋅ --> x = φ φ --> ( g ) x {\displaystyle 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:
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 {\displaystyle A,B} are objects in some category, then the set Iso --> ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} of all A → → --> ∼ ∼ --> B {\displaystyle A\mathrel {\overset {\sim }{\to }} B} is a left Aut --> ( B ) {\displaystyle \operatorname {Aut} (B)} -torsor. In practical terms, this says that a different choice of a base point of Iso --> ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} differs unambiguously by an element of Aut --> ( B ) {\displaystyle \operatorname {Aut} (B)} , or that each choice of a base point is precisely a choice of a trivialization of the torsor.
If X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are objects in categories C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} , and if F : C 1 → → --> C 2 {\displaystyle F:C_{1}\to C_{2}} is a functor mapping X 1 {\displaystyle X_{1}} to X 2 {\displaystyle X_{2}} , then F {\displaystyle F} induces a group homomorphism Aut --> ( X 1 ) → → --> Aut --> ( X 2 ) {\displaystyle \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 → → --> C {\displaystyle F:G\to C} , C a category, is called an action or a representation of G on the object F ( ∗ ∗ --> ) {\displaystyle F(*)} , or the objects F ( Obj --> ( G ) ) {\displaystyle F(\operatorname {Obj} (G))} . Those objects are then said to be G {\displaystyle G} -objects (as they are acted by G {\displaystyle G} ); cf. S {\displaystyle \mathbb {S} } -object. If C {\displaystyle C} is a module category like the category of finite-dimensional vector spaces, then G {\displaystyle G} -objects are also called G {\displaystyle G} -modules.
Let M {\displaystyle 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 → → --> M {\displaystyle M\to M} that preserve the algebraic structure: they form a vector subspace End alg --> ( M ) {\displaystyle \operatorname {End} _{\text{alg}}(M)} of End --> ( M ) {\displaystyle \operatorname {End} (M)} . The unit group of End alg --> ( M ) {\displaystyle \operatorname {End} _{\text{alg}}(M)} is the automorphism group Aut --> ( M ) {\displaystyle \operatorname {Aut} (M)} . When a basis on M is chosen, End --> ( M ) {\displaystyle \operatorname {End} (M)} is the space of square matrices and End alg --> ( M ) {\displaystyle \operatorname {End} _{\text{alg}}(M)} is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence, Aut --> ( M ) {\displaystyle \operatorname {Aut} (M)} 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 M ⊗ ⊗ --> R → → --> M ⊗ ⊗ --> R {\displaystyle M\otimes R\to M\otimes R} preserving the algebraic structure: denote it by End alg --> ( M ⊗ ⊗ --> R ) {\displaystyle \operatorname {End} _{\text{alg}}(M\otimes R)} . Then the unit group of the matrix ring End alg --> ( M ⊗ ⊗ --> R ) {\displaystyle \operatorname {End} _{\text{alg}}(M\otimes R)} over R is the automorphism group Aut --> ( M ⊗ ⊗ --> R ) {\displaystyle \operatorname {Aut} (M\otimes R)} and R ↦ ↦ --> Aut --> ( M ⊗ ⊗ --> R ) {\displaystyle 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 Aut --> ( M ) {\displaystyle \operatorname {Aut} (M)} .
In general, however, an automorphism group functor may not be represented by a scheme.
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