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 under composition of morphisms. 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.
Examples
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
, the multiplicative group of integers modulo n, 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][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 and conversely. 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 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.