# Automorphism group

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

- 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 , its automorphism group is the group consisting of field automorphisms of
*L*that fix*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

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

- Outer automorphism group
- Level structure, a trick to kill an automorphism group
- Holonomy group

## References

- Dummit & Foote, § 2.3. Exercise 26.
- Hartshorne, Ch. II, Example 7.1.1.
- Hochschild, G. (1952). "The Automorphism Group of a Lie Group".
*Transactions of the American Mathematical Society*.**72**(2): 209–216. JSTOR 1990752. - (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. - Milnor, Lemma 3.2.
- 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.