Get Transformation Group essential facts below. View Videos or join the Transformation Group discussion. Add Transformation Group to your PopFlock.com topic list for future reference or share this resource on social media.
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. 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).
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 . 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 fixesK: it is better known as the Galois group of .
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 .
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
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: 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.
^Hochschild, G. (1952). "The Automorphism Group of a Lie Group". Transactions of the American Mathematical Society. 72 (2): 209-216. JSTOR1990752.
^(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.