Automorphism group

{{short description|Mathematical group formed from the automorphisms of an object}}

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 $\operatorname{Aut}(X)$ 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 $\operatorname{PGL}_n(k).$ {{harvnb|Hartshorne|1977|loc=Ch. II, Example 7.1.1.}}
The automorphism group $G$ of a finite cyclic group of order n is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$ , the multiplicative group of integers modulo n, with the isomorphism given by $\overline{a} \mapsto \sigma_a \in G, \, \sigma_a(x) = x^a$ .{{harvnb|Dummit|Foote|2004|loc=§ 2.3. Exercise 26.}} 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 $\mathfrak{g}$ .{{Cite journal |jstor = 1990752|title = The Automorphism Group of a Lie Group|journal = Transactions of the American Mathematical Society|volume = 72|issue = 2|pages = 209–216|last1 = Hochschild|first1 = G.|year = 1952| doi=10.2307/1990752 }}{{sfn|Fulton|Harris|1991|loc=Exercise 8.28}}{{efn|First, if G is simply connected, the automorphism group of G is that of $\mathfrak{g}$ . Second, every connected Lie group is of the form $\widetilde{G}/C$ where $\widetilde{G}$ is a simply connected Lie group and C is a central subgroup and the automorphism group of G is the automorphism group of $G$ 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.}}

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 $G \to \operatorname{Aut}(X), \, g \mapsto \sigma_g, \, \sigma_g(x) = g \cdot x$ , 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.{{harvnb|Milnor|1971|loc=Lemma 3.2.}}

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 $F: G \to C$ , 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:{{harvnb|Waterhouse|2012|loc=§ 7.6.}} 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.

Notes

Citations

References

{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=Wiley | year=2004 | edition=3rd | isbn=978-0-471-43334-7}}
{{Fulton-Harris}}
{{Hartshorne AG}}
{{Cite book | last=Milnor | first=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=Princeton University Press | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 |url=https://books.google.com/books?id=zVYVhjQX4lYC |isbn=9780691081014}}
{{cite book |author-link=William C. Waterhouse |first=William C. |last=Waterhouse |title=Introduction to Affine Group Schemes |publisher=Springer Verlag |series=Graduate Texts in Mathematics |volume=66 |orig-year=1979 |year=2012 |isbn=9781461262176 |url=https://books.google.com/books?id=SpfwBwAAQBAJ}}

External links

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

Category:Group automorphisms