 Automorphism Group
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Automorphism Group

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the general linear group of X, the group of invertible linear transformations from X to itself.

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is called a transformation group (especially in old literature).

## Examples

• The automorphism group of a set X is precisely the symmetric group of X.
• A group homomorphism to the automorphism group of a set X amounts to a group action on X: 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 $g\cdot x=\varphi (g)x$ .
• Let $A,B$ be two finite sets of the same cardinality and $\operatorname {Iso} (A,B)$ the set of all bijections $A\ {\overset {\sim }{\to }}\ B$ . 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).
• The automorphism group $G$ of a finite cyclic group of order n is isomorphic to $(\mathbb {Z} /n\mathbb {Z} )^{*}$ with the isomorphism given by ${\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}$ . In particular, $G$ is an abelian group.
• Given a field extension $L/K$ , its automorphism group is the group consisting of field automorphisms of L that fix K: it is better known as the Galois group of $L/K$ .
• The automorphism group of the projective n-space over a field k is the projective linear group $\operatorname {PGL} _{n}(k).$ • 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 ${\mathfrak {g}}$ .
• 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 $A{\overset {\sim }{\to }}B$ 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 $G\to C$ , 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.