# Automorphism group

In mathematics, the automorphism group in one of its most general forms is defined in the context of category theory. In category theory, the automorphism group of an object X is the group consisting of automorphisms of X. The most famous example is the ${\displaystyle \operatorname {Aut} (G)}$, which is the group of automorphisms on a group like ${\displaystyle G}$, another one is the general linear group: 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.

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 ${\displaystyle G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x}$ , and, conversely, each homomorphism ${\displaystyle \varphi :G\to \operatorname {Aut} (X)}$  defines an action by ${\displaystyle g\cdot x=\varphi (g)x}$ .
• Let ${\displaystyle A,B}$  be two finite sets of the same cardinality and ${\displaystyle \operatorname {Iso} (A,B)}$  the set of all bijections ${\displaystyle A\mathrel {\overset {\sim }{\to }} B}$ . Then ${\displaystyle \operatorname {Aut} (B)}$ , which is a symmetric group (see above), acts on ${\displaystyle \operatorname {Iso} (A,B)}$  from the left freely and transitively; that is to say, ${\displaystyle \operatorname {Iso} (A,B)}$  is a torsor for ${\displaystyle \operatorname {Aut} (B)}$  (cf. #In category theory).
• The automorphism group ${\displaystyle G}$  of a finite cyclic group of order n is isomorphic to ${\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{*}}$  with the isomorphism given by ${\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}}$ .[1] In particular, ${\displaystyle G}$  is an abelian group.
• The automorphism group of a field extension ${\displaystyle L/K}$  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 ${\displaystyle \operatorname {PGL} _{n}(k).}$ [2]
• The automorphism group of a finite-dimensional real Lie algebra ${\displaystyle {\mathfrak {g}}}$  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 ${\displaystyle {\mathfrak {g}}}$ , then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of ${\displaystyle {\mathfrak {g}}}$ .[3][4][a]
• Let P be a finitely generated projective module over a ring R. Then there is an embedding ${\displaystyle \operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)}$ , unique up to inner automorphisms.[5]

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

If ${\displaystyle X_{1}}$  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 a functor mapping ${\displaystyle X_{1}}$  to ${\displaystyle X_{2}}$ , then ${\displaystyle F}$  induces a group homomorphism ${\displaystyle \operatorname {Aut} (X_{1})\to \operatorname {Aut} (X_{2})}$ , 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 ${\displaystyle G\to C}$ , C a category, is called an action or a representation of G on the object ${\displaystyle F(*)}$ , or the objects ${\displaystyle F(\operatorname {Obj} (G))}$ . 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.

## Automorphism group functor

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

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

## Notes

1. ^ First, if G is simply connected, the automorphism group of G is that of ${\displaystyle {\mathfrak {g}}}$ . Second, every connected Lie group is of the form ${\displaystyle {\widetilde {G}}/C}$  where ${\displaystyle {\widetilde {G}}}$  is a simply connected Lie group and C is a central subgroup and the automorphism group of G is the automorphism group of ${\displaystyle G}$  that preserves C. Third, by convention, a Lie group is second countable and has at most coutably many connected components; thus, the general case reduces to the connected case.

## Citations

1. ^ Dummit & Foote 2004, § 2.3. Exercise 26.
2. ^ Hartshorne 1977, Ch. II, Example 7.1.1.
3. ^ Hochschild, G. (1952). "The Automorphism Group of a Lie Group". Transactions of the American Mathematical Society. 72 (2): 209–216. JSTOR 1990752.
4. ^ Fulton & Harris 1991, Exercise 8.28.
5. ^ Milnor 1971, Lemma 3.2.
6. ^ Waterhouse 2012, § 7.6.