# Automorphism group

(Redirected from Transformation 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 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 $\operatorname {Aut} (X)$ 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.

## Examples

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 $(\mathbb {Z} /n\mathbb {Z} )^{\times }$  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 ${\mathfrak {g}}$ .[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 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 $g\cdot x=\varphi (g)x$ . 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 $G\to C$ , 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.