Automorphism group

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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 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 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.

ExamplesEdit

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   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  [1]
  • The automorphism group   of a finite cyclic group of order n is isomorphic to   with the isomorphism given by  .[2] In particular,   is an abelian group.
  • The automorphism group of a finite-dimensional real Lie algebra   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  , then the automorphism group of G has a structure of a Lie group induced from that on the automorphism group of  .[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 amounts to a group action on X. Indeed, each left G-action on a set X determines  , and, conversely, each homomorphism   defines an action by  . 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   be two finite sets of the same cardinality and   the set of all bijections  . Then  , which is a symmetric group (see above), acts on   from the left freely and transitively; that is to say,   is a torsor for   (cf. #In category theory).
  • Let P be a finitely generated projective module over a ring R. Then there is an embedding  , unique up to inner automorphisms.[5]

In category theoryEdit

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   are objects in some category, then the set   of all   is a left  -torsor. In practical terms, this says that a different choice of a base point of   differs unambiguously by an element of  , or that each choice of a base point is precisely a choice of a trivialization of the torsor.

If   and   are objects in categories   and  , and if   is a functor mapping   to  , then   induces a group homomorphism  , 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  , C a category, is called an action or a representation of G on the object  , or the objects  . Those objects are then said to be  -objects (as they are acted by  ); cf.  -object. If   is a module category like the category of finite-dimensional vector spaces, then  -objects are also called  -modules.

Automorphism group functorEdit

Let   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   that preserve the algebraic structure: they form a vector subspace   of  . The unit group of   is the automorphism group  . When a basis on M is chosen,   is the space of square matrices and   is the zero set of some polynomial equations, and the invertibility is again described by polynomials. Hence,   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   preserving the algebraic structure: denote it by  . Then the unit group of the matrix ring   over R is the automorphism group   and   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  .

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

See alsoEdit

NotesEdit

  1. ^ First, if G is simply connected, the automorphism group of G is that of  . Second, every connected Lie group is of the form   where   is a simply connected Lie group and C is a central subgroup and the automorphism group of G is the automorphism group of   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.

CitationsEdit

  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. JSTOR 1990752.
  4. ^ Fulton & Harris 1991, Exercise 8.28.
  5. ^ Milnor 1971, Lemma 3.2.
  6. ^ Waterhouse 2012, § 7.6.

ReferencesEdit

External linksEdit