Automorphism group

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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.

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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] [lower-alpha 1]
• 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.

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