Automorphism group

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. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the general linear group of X, 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).

Contents

• 
  1 Examples
  


• 
  2 In category theory
  


• 
  3 Automorphism group functor
  


• 
  4 See also
  


• 
  5 References
  


• 
  6 External links
  

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 G→Aut⁡(X),g↦σg,σg(x)=g⋅x{displaystyle Gto operatorname {Aut} (X),,gmapsto sigma _{g},,sigma _{g}(x)=gcdot x}, and, conversely, each homomorphism φ:G→Aut⁡(X){displaystyle varphi :Gto operatorname {Aut} (X)} defines an action by g⋅x=φ(g)x{displaystyle gcdot x=varphi (g)x}.


• Let A,B{displaystyle A,B} be two finite sets of the same cardinality and Iso⁡(A,B){displaystyle operatorname {Iso} (A,B)} the set of all bijections A→∼B{displaystyle A{overset {sim }{to }}B}. Then Aut⁡(B){displaystyle operatorname {Aut} (B)}, which is a symmetric group (see above), acts on 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)} is a torsor for Aut⁡(B){displaystyle operatorname {Aut} (B)} (cf. #In category theory).


• The automorphism group G{displaystyle G} of a finite cyclic group of order n is isomorphic to (Z/nZ)∗{displaystyle (mathbb {Z} /nmathbb {Z} )^{*}} with the isomorphism given by a¯↦σa∈G,σa(x)=xa{displaystyle {overline {a}}mapsto sigma _{a}in G,,sigma _{a}(x)=x^{a}}.^[1] In particular, G{displaystyle G} is an abelian group.


• Given a field extension L/K{displaystyle L/K}, the automorphism group of it is the group consisting of field automorphisms of L that fixes K: it is better known as the Galois group of L/K{displaystyle L/K}.


• The automorphism group of the projective n-space over a field k is the projective linear group PGLn⁡(k).{displaystyle operatorname {PGL} _{n}(k).}^[2]
  


• The automorphism group of a finite-dimensional real Lie algebra 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}}}, 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}}}.^[3][4]
  


• Let P be a finitely generated projective module over a ring R. Then there is an embedding Aut⁡(P)↪GLn⁡(R){displaystyle operatorname {Aut} (P)hookrightarrow operatorname {GL} _{n}(R)}, unique up to inner automorphisms.^[5]
  

In category theory

Automorphism groups appear very natural 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 example, see PROP.)

If A,B{displaystyle A,B} are objects in some category, then the set Iso⁡(A,B){displaystyle operatorname {Iso} (A,B)} of all A→∼B{displaystyle A{overset {sim }{to }}B} is a left 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)} differs unambiguously by an element of 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 Xi,i=1,2{displaystyle X_{i},i=1,2} are objects in categories Ci{displaystyle C_{i}} and if F:C1→C2{displaystyle F:C_{1}to C_{2}} is a functor that maps X1{displaystyle X_{1}} to X2{displaystyle X_{2}}, then the functor F{displaystyle F} induces a group homomorphism Aut⁡(X1)→Aut⁡(X2){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 G→C{displaystyle Gto C}, C a category, is called an action or a representation of G on the object F(∗){displaystyle F(*)}, or the objects F(Obj⁡(G)){displaystyle F(operatorname {Obj} (G))}. Those objects are then said to be G{displaystyle G}-objects (as they are acted by G{displaystyle G}); cf. S{displaystyle mathbb {S} }-object. If C{displaystyle C} is a module category like the category of finite-dimensional vector spaces, then G{displaystyle G}-objects are also called G{displaystyle G}-modules.

Automorphism group functor

Let 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 Mto M} that preserve the algebraic structure: they form a vector subspace Endalg⁡(M){displaystyle operatorname {End} _{text{alg}}(M)} of End⁡(M){displaystyle operatorname {End} (M)}. The unit group of Endalg⁡(M){displaystyle operatorname {End} _{text{alg}}(M)} is the automorphism group Aut⁡(M){displaystyle operatorname {Aut} (M)}. When a basis on M is chosen, End⁡(M){displaystyle operatorname {End} (M)} is the space of square matrices and Endalg⁡(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)} 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 Motimes Rto Motimes R} preserving the algebraic structure: denote it by Endalg⁡(M⊗R){displaystyle operatorname {End} _{text{alg}}(Motimes R)}. Then the unit group of the matrix ring Endalg⁡(M⊗R){displaystyle operatorname {End} _{text{alg}}(Motimes R)} over R is the automorphism group Aut⁡(M⊗R){displaystyle operatorname {Aut} (Motimes R)} and R↦Aut⁡(M⊗R){displaystyle Rmapsto operatorname {Aut} (Motimes 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)}.

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

See also

• Outer automorphism group


• 
  Level structure, a trick to kill an automorphism group


• Holonomy group

References

1. 
   ^ Dummit & Foote, § 2.3. Exercise 26.
   


2. 
   ^ Hartshorne, 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..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
   


4. 
   ^ (following Fulton–Harris, Exercise 8.28.) First, if G is simply connected, the automorphism group of G is that of g{displaystyle {mathfrak {g}}}. Second, every connected Lie group is of the form G~/C{displaystyle {widetilde {G}}/C} where 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} 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.
   


5. 
   ^ Milnor, Lemma 3.2.
   


6. 
   ^ Waterhouse, § 7.6.
   

• 
  Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 978-0-471-43334-7.
  


• 
  Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
  


• 
  Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
  


• 
  Milnor, John Willard (1971), Introduction to algebraic K-theory, Annals of Mathematics Studies, 72, Princeton, NJ: Princeton University Press, MR 0349811, Zbl 0237.18005
  


• 
  William C. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics vol. 66, Springer Verlag New York, 1979.

External links

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

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