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 ${\displaystyle \operatorname {Aut} (X)}${\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:

• The automorphism group of a field extension ${\displaystyle L/K}${\displaystyle 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 ${\displaystyle \operatorname {PGL} _{n}(k).}${\displaystyle \operatorname {PGL} _{n}(k).}^[1]
• The automorphism group ${\displaystyle G}${\displaystyle G} of a finite cyclic group of order n is isomorphic to ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}, the multiplicative group of integers modulo n, with the isomorphism given by ${\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,,円\sigma _{a}(x)=x^{a}}${\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,,円\sigma _{a}(x)=x^{a}}.^[2] In particular, ${\displaystyle G}${\displaystyle G} is an abelian group.
• The automorphism group of a finite-dimensional real Lie algebra ${\displaystyle {\mathfrak {g}}}${\displaystyle {\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 ${\displaystyle {\mathfrak {g}}}${\displaystyle {\mathfrak {g}}}, then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of ${\displaystyle {\mathfrak {g}}}${\displaystyle {\mathfrak {g}}}.^{[3] [4] [a]}

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 ${\displaystyle G\to \operatorname {Aut} (X),,円g\mapsto \sigma _{g},,円\sigma _{g}(x)=g\cdot x}${\displaystyle G\to \operatorname {Aut} (X),,円g\mapsto \sigma _{g},,円\sigma _{g}(x)=g\cdot x}, and, conversely, each homomorphism ${\displaystyle \varphi :G\to \operatorname {Aut} (X)}${\displaystyle \varphi :G\to \operatorname {Aut} (X)} defines an action by ${\displaystyle g\cdot x=\varphi (g)x}${\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:

• Let ${\displaystyle A,B}${\displaystyle A,B} be two finite sets of the same cardinality and ${\displaystyle \operatorname {Iso} (A,B)}${\displaystyle \operatorname {Iso} (A,B)} the set of all bijections ${\displaystyle A\mathrel {\overset {\sim }{\to }} B}${\displaystyle A\mathrel {\overset {\sim }{\to }} B}. Then ${\displaystyle \operatorname {Aut} (B)}${\displaystyle \operatorname {Aut} (B)}, which is a symmetric group (see above), acts on ${\displaystyle \operatorname {Iso} (A,B)}${\displaystyle \operatorname {Iso} (A,B)} from the left freely and transitively; that is to say, ${\displaystyle \operatorname {Iso} (A,B)}${\displaystyle \operatorname {Iso} (A,B)} is a torsor for ${\displaystyle \operatorname {Aut} (B)}${\displaystyle \operatorname {Aut} (B)} (cf. #In category theory).
• Let P be a finitely generated projective module over a ring R. Then there is an embedding ${\displaystyle \operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)}${\displaystyle \operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)}, unique up to inner automorphisms.^[5]

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

If ${\displaystyle X_{1}}${\displaystyle X_{1}} and ${\displaystyle X_{2}}${\displaystyle X_{2}} are objects in categories ${\displaystyle C_{1}}${\displaystyle C_{1}} and ${\displaystyle C_{2}}${\displaystyle C_{2}}, and if ${\displaystyle F:C_{1}\to C_{2}}${\displaystyle F:C_{1}\to C_{2}} is a functor mapping ${\displaystyle X_{1}}${\displaystyle X_{1}} to ${\displaystyle X_{2}}${\displaystyle X_{2}}, then ${\displaystyle F}${\displaystyle F} induces a group homomorphism ${\displaystyle \operatorname {Aut} (X_{1})\to \operatorname {Aut} (X_{2})}${\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 ${\displaystyle F:G\to C}${\displaystyle F:G\to C}, C a category, is called an action or a representation of G on the object ${\displaystyle F(*)}${\displaystyle F(*)}, or the objects ${\displaystyle F(\operatorname {Obj} (G))}${\displaystyle F(\operatorname {Obj} (G))}. Those objects are then said to be ${\displaystyle G}${\displaystyle G}-objects (as they are acted by ${\displaystyle G}${\displaystyle G}); cf. ${\displaystyle \mathbb {S} }${\displaystyle \mathbb {S} }-object. If ${\displaystyle C}${\displaystyle C} is a module category like the category of finite-dimensional vector spaces, then ${\displaystyle G}${\displaystyle G}-objects are also called ${\displaystyle G}${\displaystyle G}-modules.

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