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 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).

Examples[edit]

  • 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 , and, conversely, each homomorphism defines an action by .
  • 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).
  • The automorphism group of a finite cyclic group of order n is isomorphic to with the isomorphism given by .[1] In particular, is an abelian group.
  • Given a field extension , 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 .
  • The automorphism group of the projective n-space over a field k is the projective linear group [2]
  • 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]
  • 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 theory[edit]

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 example, 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 are objects in categories and if is a functor that maps to , then the functor 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 functor[edit]

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 also[edit]

References[edit]

  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.
  4. ^ (following Fulton–Harris, Exercise 8.28.) 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.
  5. ^ Milnor, Lemma 3.2.
  6. ^ Waterhouse, § 7.6.

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