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 , which is the group of automorphisms on a group like , 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.


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).


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 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 functor

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


  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.


  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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