Inmathematics, theautomorphism group of an objectX is thegroup consisting ofautomorphisms ofX undercomposition ofmorphisms. For example, ifX is afinite-dimensionalvector space, then the automorphism group ofX is the group of invertiblelinear transformations fromX to itself (thegeneral linear group ofX). If insteadX is a group, then its automorphism group${\displaystyle \mathrm{Aut}(X)}$ is the group consisting of allgroup automorphisms ofX.

Especially in geometric contexts, an automorphism group is also called asymmetry group. A subgroup of an automorphism group is sometimes called atransformation group.

Automorphism groups are studied in a general way in the field ofcategory theory.

IfX is aset with no additional structure, then any bijection fromX to itself is an automorphism, and hence the automorphism group ofX in this case is precisely thesymmetric group ofX. If the setX 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 onX. Some examples of this include the following:

• The automorphism group of afield extension${\displaystyle L/K}$ is the group consisting of field automorphisms ofL thatfixK. If the field extension isGalois, the automorphism group is called theGalois group of the field extension.
• The automorphism group of theprojectiven-space over afieldk is theprojective linear group${\displaystyle {\mathrm{PGL}}_n(k).}$^[1]
• The automorphism group${\displaystyle G}$ of a finitecyclic group ofordern isisomorphic to${\displaystyle (\mathbb{Z}/n\mathbb{Z}{)}^{\times }}$, themultiplicative group of integers modulon, with the isomorphism given by${\displaystyle \overline{a}\mapsto {\sigma }_a\in G,{\sigma }_a(x)=x^a}$.^[2] In particular,${\displaystyle G}$ is anabelian group.
• The automorphism group of a finite-dimensional realLie algebra${\displaystyle \mathfrak{g}}$ has the structure of a (real)Lie group (in fact, it is even alinear algebraic group: seebelow). IfG is a Lie group with Lie algebra${\displaystyle \mathfrak{g}}$, then the automorphism group ofG has a structure of a Lie group induced from that on the automorphism group of${\displaystyle \mathfrak{g}}$.^[3][4][a]

IfG is a groupacting on a setX, the action amounts to agroup homomorphism fromG to the automorphism group ofX and conversely. Indeed, each leftG-action on a setX determines${\displaystyle G\to \mathrm{Aut}(X),g\mapsto {\sigma }_g,{\sigma }_g(x)=g\cdot x}$, and, conversely, each homomorphism${\displaystyle \phi :G\to \mathrm{Aut}(X)}$ defines an action by${\displaystyle g\cdot x=\phi (g)x}$. This extends to the case when the setX has more structure than just a set. For example, ifX is a vector space, then a group action ofG onX is agroup representation of the groupG, representingG as a group of linear transformations (automorphisms) ofX; these representations are the main object of study in the field ofrepresentation theory.

Here are some other facts about automorphism groups:

• Let${\displaystyle A,B}$ be two finite sets of the samecardinality and${\displaystyle \mathrm{Iso}(A,B)}$ the set of allbijections${\displaystyle A\stackrel{\sim }{\to }B}$. Then${\displaystyle \mathrm{Aut}(B)}$, which is a symmetric group (see above), acts on${\displaystyle \mathrm{Iso}(A,B)}$ from the leftfreely andtransitively; that is to say,${\displaystyle \mathrm{Iso}(A,B)}$ is atorsor for${\displaystyle \mathrm{Aut}(B)}$ (cf.#In category theory).
• LetP be afinitely generatedprojective module over aringR. Then there is anembedding${\displaystyle \mathrm{Aut}(P)\hookrightarrow {\mathrm{GL}}_n(R)}$, unique up toinner automorphisms.^[5]

Automorphism groups appear very naturally incategory theory.

IfX is anobject in a category, then the automorphism group ofX is the group consisting of all the invertiblemorphisms fromX to itself. It is theunit group of theendomorphism monoid ofX. (For some examples, seePROP.)

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

If${\displaystyle X_1}$ and${\displaystyle X_2}$ are objects in categories${\displaystyle C_1}$ and${\displaystyle C_2}$, and if${\displaystyle F:C_1\to C_2}$ is afunctor mapping${\displaystyle X_1}$ to${\displaystyle X_2}$, then${\displaystyle F}$ induces a group homomorphism${\displaystyle \mathrm{Aut}(X_1)\to \mathrm{Aut}(X_2)}$, as it maps invertible morphisms to invertible morphisms.

In particular, ifG is a group viewed as acategory with a single object * or, more generally, ifG is a groupoid, then each functor${\displaystyle F:G\to C}$,C a category, is called an action or a representation ofG on the object${\displaystyle F(\ast )}$, or the objects${\displaystyle F(\mathrm{Obj}(G))}$. Those objects are then said to be${\displaystyle G}$-objects (as they are acted by${\displaystyle G}$); cf.${\displaystyle \mathbb{S}}$-object. If${\displaystyle C}$ is a module category like the category of finite-dimensional vector spaces, then${\displaystyle G}$-objects are also called${\displaystyle G}$-modules.

Let${\displaystyle M}$ be a finite-dimensional vector space over a fieldk that is equipped with some algebraic structure (that is,M is a finite-dimensionalalgebra overk). It can be, for example, anassociative algebra or aLie algebra.

Now, considerk-linear maps${\displaystyle M\to M}$ that preserve the algebraic structure: they form avector subspace${\displaystyle {\mathrm{End}}_{\text{alg}}(M)}$ of${\displaystyle \mathrm{End}(M)}$. The unit group of${\displaystyle {\mathrm{End}}_{\text{alg}}(M)}$ is the automorphism group${\displaystyle \mathrm{Aut}(M)}$. When a basis onM is chosen,${\displaystyle \mathrm{End}(M)}$ is the space ofsquare matrices and${\displaystyle {\mathrm{End}}_{\text{alg}}(M)}$ is the zero set of somepolynomial equations, and the invertibility is again described by polynomials. Hence,${\displaystyle \mathrm{Aut}(M)}$ is alinear algebraic group overk.

Now base extensions applied to the above discussion determines a functor:^[6] namely, for eachcommutative ringR overk, consider theR-linear maps${\displaystyle M\otimes R\to M\otimes R}$ preserving the algebraic structure: denote it by${\displaystyle {\mathrm{End}}_{\text{alg}}(M\otimes R)}$. Then the unit group of the matrix ring${\displaystyle {\mathrm{End}}_{\text{alg}}(M\otimes R)}$ overR is the automorphism group${\displaystyle \mathrm{Aut}(M\otimes R)}$ and${\displaystyle R\mapsto \mathrm{Aut}(M\otimes R)}$ is agroup functor: a functor from thecategory of commutative rings overk to thecategory of groups. Even better, it is represented by a scheme (since the automorphism groups are defined by polynomials): this scheme is called theautomorphism group scheme and is denoted by${\displaystyle \mathrm{Aut}(M)}$.

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

• Outer automorphism group
• Level structure, a technique to remove an automorphism group
• Holonomy group

• https://mathoverflow.net/questions/55042/automorphism-group-of-a-scheme

• Group automorphisms

• Articles with short description
• Short description matches Wikidata