Automorphism group

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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 Aut ⁡ ( X ) {\displaystyle \operatorname {Aut} (X)} {\displaystyle \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 {\displaystyle L/K} {\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 PGL n ⁡ ( k ) . {\displaystyle \operatorname {PGL} _{n}(k).} {\displaystyle \operatorname {PGL} _{n}(k).}[1]
  • The automorphism group G {\displaystyle G} {\displaystyle G} of a finite cyclic group of order n is isomorphic to ( Z / n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }}, the multiplicative group of integers modulo n, with the isomorphism given by a ¯ ↦ σ a ∈ G , σ a ( x ) = x a {\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}} {\displaystyle {\overline {a}}\mapsto \sigma _{a}\in G,\,\sigma _{a}(x)=x^{a}}.[2] In particular, G {\displaystyle G} {\displaystyle G} is an abelian group.
  • The automorphism group of a finite-dimensional real Lie algebra g {\displaystyle {\mathfrak {g}}} {\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 g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}, then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}.[3][4][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 and conversely. Indeed, each left G-action on a set X determines G → Aut ⁡ ( X ) , g ↦ σ g , σ g ( x ) = g ⋅ x {\displaystyle G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x} {\displaystyle G\to \operatorname {Aut} (X),\,g\mapsto \sigma _{g},\,\sigma _{g}(x)=g\cdot x}, and, conversely, each homomorphism φ : G → Aut ⁡ ( X ) {\displaystyle \varphi :G\to \operatorname {Aut} (X)} {\displaystyle \varphi :G\to \operatorname {Aut} (X)} defines an action by g ⋅ x = φ ( g ) x {\displaystyle g\cdot x=\varphi (g)x} {\displaystyle 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 {\displaystyle A,B} {\displaystyle A,B} be two finite sets of the same cardinality and Iso ⁡ ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} {\displaystyle \operatorname {Iso} (A,B)} the set of all bijections A → ∼ B {\displaystyle A\mathrel {\overset {\sim }{\to }} B} {\displaystyle A\mathrel {\overset {\sim }{\to }} B}. Then Aut ⁡ ( B ) {\displaystyle \operatorname {Aut} (B)} {\displaystyle \operatorname {Aut} (B)}, which is a symmetric group (see above), acts on Iso ⁡ ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} {\displaystyle \operatorname {Iso} (A,B)} from the left freely and transitively; that is to say, Iso ⁡ ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} {\displaystyle \operatorname {Iso} (A,B)} is a torsor for Aut ⁡ ( B ) {\displaystyle \operatorname {Aut} (B)} {\displaystyle \operatorname {Aut} (B)} (cf. #In category theory).
  • Let P be a finitely generated projective module over a ring R. Then there is an embedding Aut ⁡ ( P ) ↪ GL n ⁡ ( R ) {\displaystyle \operatorname {Aut} (P)\hookrightarrow \operatorname {GL} _{n}(R)} {\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 A , B {\displaystyle A,B} {\displaystyle A,B} are objects in some category, then the set Iso ⁡ ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} {\displaystyle \operatorname {Iso} (A,B)} of all A → ∼ B {\displaystyle A\mathrel {\overset {\sim }{\to }} B} {\displaystyle A\mathrel {\overset {\sim }{\to }} B} is a left Aut ⁡ ( B ) {\displaystyle \operatorname {Aut} (B)} {\displaystyle \operatorname {Aut} (B)}-torsor. In practical terms, this says that a different choice of a base point of Iso ⁡ ( A , B ) {\displaystyle \operatorname {Iso} (A,B)} {\displaystyle \operatorname {Iso} (A,B)} differs unambiguously by an element of Aut ⁡ ( B ) {\displaystyle \operatorname {Aut} (B)} {\displaystyle \operatorname {Aut} (B)}, or that each choice of a base point is precisely a choice of a trivialization of the torsor.

If X 1 {\displaystyle X_{1}} {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} {\displaystyle X_{2}} are objects in categories C 1 {\displaystyle C_{1}} {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} {\displaystyle C_{2}}, and if F : C 1 → C 2 {\displaystyle F:C_{1}\to C_{2}} {\displaystyle F:C_{1}\to C_{2}} is a functor mapping X 1 {\displaystyle X_{1}} {\displaystyle X_{1}} to X 2 {\displaystyle X_{2}} {\displaystyle X_{2}}, then F {\displaystyle F} {\displaystyle F} induces a group homomorphism Aut ⁡ ( X 1 ) → Aut ⁡ ( X 2 ) {\displaystyle \operatorname {Aut} (X_{1})\to \operatorname {Aut} (X_{2})} {\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 F : G → C {\displaystyle F:G\to C} {\displaystyle F:G\to C}, C a category, is called an action or a representation of G on the object F ( ∗ ) {\displaystyle F(*)} {\displaystyle F(*)}, or the objects F ( Obj ⁡ ( G ) ) {\displaystyle F(\operatorname {Obj} (G))} {\displaystyle F(\operatorname {Obj} (G))}. Those objects are then said to be G {\displaystyle G} {\displaystyle G}-objects (as they are acted by G {\displaystyle G} {\displaystyle G}); cf. S {\displaystyle \mathbb {S} } {\displaystyle \mathbb {S} }-object. If C {\displaystyle C} {\displaystyle C} is a module category like the category of finite-dimensional vector spaces, then G {\displaystyle G} {\displaystyle G}-objects are also called G {\displaystyle G} {\displaystyle G}-modules.

Automorphism group functor

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

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

See also

Notes

  1. First, if G is simply connected, the automorphism group of G is that of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}. Second, every connected Lie group is of the form G ~ / C {\displaystyle {\widetilde {G}}/C} {\displaystyle {\widetilde {G}}/C} where G ~ {\displaystyle {\widetilde {G}}} {\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 G {\displaystyle G} {\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. Hartshorne 1977, Ch. II, Example 7.1.1.
  2. Dummit & Foote 2004, § 2.3. Exercise 26.
  3. Hochschild, G. (1952). "The Automorphism Group of a Lie Group". Transactions of the American Mathematical Society. 72 (2): 209–216. doi:10.2307/1990752. JSTOR 1990752.
  4. Fulton & Harris 1991, Exercise 8.28.
  5. Milnor 1971, Lemma 3.2.
  6. Waterhouse 2012, § 7.6.

References