TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, the automorphism group of an object ''X'' is the
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
consisting of
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of ...
s of ''X''. For example, if ''X'' is a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disting ...
vector space#REDIRECT Vector space#REDIRECT Vector space {{Redirect category shell, 1= {{R for alternate capitalisation ...
{{Redirect category shell, 1= {{R for alternate capitalisation ...
, then the automorphism group of ''X'' is the
general linear group A general officer is an officer of high rank in the armies, and in some nations' air forces, space forces, or marines. The term ''general'' is used in two ways: as the generic title for all grades of general officer and as a specific rank. It ...
of ''X'', the group of invertible
linear transformations In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \rightarrow W between two vector spaces that preserves the operations of vector addi ...
from ''X'' to itself. Especially in geometric contexts, an automorphism group is also called a
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient s ...
. 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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group S_n de ...
of ''X''. *A
group homomorphismImage:Group homomorphism ver.2.svg, 250px, Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the Kernel_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N. In mathema ...
to the automorphism group of a set ''X'' amounts to a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism gr ...
on ''X'': indeed, each left ''G''-action on a set ''X'' determines $G \to \operatorname\left(X\right), \, g \mapsto \sigma_g, \, \sigma_g\left(x\right) = g \cdot x$, and, conversely, each homomorphism $\varphi: G \to \operatorname\left(X\right)$ defines an action by $g \cdot x = \varphi\left(g\right)x$. *Let $A, B$ be two finite sets of the same
cardinality In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
and $\operatorname\left(A, B\right)$ the set of all
bijection In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ...

s $A \mathrel B$. Then $\operatorname\left(B\right)$, which is a symmetric group (see above), acts on $\operatorname\left(A, B\right)$ from the left freely and transitively; that is to say, $\operatorname\left(A, B\right)$ is a
torsor:''For the term "torsor" in algebraic geometry, see torsor (algebraic geometry).'' In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is t ...
for $\operatorname\left(B\right)$ (cf. #In category theory). *The automorphism group $G$ of a finite cyclic group of Order (group theory), order ''n'' is Group isomorphism, isomorphic to $\left(\mathbb/n\mathbb\right)^*$ with the isomorphism given by $\overline \mapsto \sigma_a \in G, \, \sigma_a\left(x\right) = x^a$. In particular, $G$ is an abelian group. *The automorphism group of a field extension $L/K$ is the group consisting of field automorphisms of ''L'' that fixed-point subring, fix ''K''. If the field extension is Galois extension, Galois, the automorphism group is called the Galois group of the field extension. *The automorphism group of the projective space, projective ''n''-space over a Field (mathematics), field ''k'' is the projective linear group $\operatorname_n\left(k\right).$ *The automorphism group of a finite-dimensional real Lie algebra $\mathfrak$ 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 $\mathfrak$, then the automorphism group of ''G'' has a structure of a Lie group induced from that on the automorphism group of $\mathfrak$. *Let ''P'' be a finitely generated module, finitely generated projective module over a Ring (mathematics), ring ''R''. Then there is an embedding $\operatorname\left(P\right) \hookrightarrow \operatorname_n\left(R\right)$, unique up to inner automorphisms.

# In category theory

Automorphism groups appear very naturally in category theory. If ''X'' is an Object (category theory), 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 (category theory), PROP.) If $A, B$ are objects in some category, then the set $\operatorname\left(A, B\right)$ of all $A \mathrel B$ is a left $\operatorname\left(B\right)$-principal homogeneous space, torsor. In practical terms, this says that a different choice of a base point of $\operatorname\left(A, B\right)$ differs unambiguously by an element of $\operatorname\left(B\right)$, or that each choice of a base point is precisely a choice of a trivialization of the torsor. If $X_1$ and $X_2$ are objects in categories $C_1$ and $C_2$, and if $F: C_1 \to C_2$ is a functor mapping $X_1$ to $X_2$, then $F$ induces a group homomorphism $\operatorname\left(X_1\right) \to \operatorname\left(X_2\right)$, as it maps invertible morphisms to invertible morphisms. In particular, if ''G'' is a group viewed as a Category (mathematics), category with a single object * or, more generally, if ''G'' is a groupoid, then each functor $G \to C$, ''C'' a category, is called an action or a representation of ''G'' on the object $F\left(*\right)$, or the objects $F\left(\operatorname\left(G\right)\right)$. Those objects are then said to be $G$-objects (as they are acted by $G$); cf. S-object, $\mathbb$-object. If $C$ is a module category like the category of finite-dimensional vector spaces, then $G$-objects are also called $G$-modules.

# Automorphism group functor

Let $M$ 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 a field, algebra over ''k''). It can be, for example, an associative algebra or a Lie algebra. Now, consider ''k''-linear maps $M \to M$ that preserve the algebraic structure: they form a vector subspace $\operatorname_\left(M\right)$ of $\operatorname\left(M\right)$. The unit group of $\operatorname_\left(M\right)$ is the automorphism group $\operatorname\left(M\right)$. When a basis on ''M'' is chosen, $\operatorname\left(M\right)$ is the space of Square matrix, square matrices and $\operatorname_\left(M\right)$ is the zero set of some Polynomial, polynomial equations, and the invertibility is again described by polynomials. Hence, $\operatorname\left(M\right)$ 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 $M \otimes R \to M \otimes R$ preserving the algebraic structure: denote it by $\operatorname_\left(M \otimes R\right)$. Then the unit group of the matrix ring $\operatorname_\left(M \otimes R\right)$ over ''R'' is the automorphism group $\operatorname\left(M \otimes R\right)$ and $R \mapsto \operatorname\left(M \otimes R\right)$ is a group functor: a functor from the Category_of_rings#Category_of_commutative_rings, 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 $\operatorname\left(M\right)$. In general, however, an automorphism group functor may not be represented by a scheme.