In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the automorphism group of an object ''X'' is the group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

consisting of automorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...

vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, then the automorphism group of ''X'' is the general linear group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

of ''X'', the group of invertible linear transformations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

from ''X'' to itself.
Especially in geometric contexts, an automorphism group is also called a symmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

. 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 thesymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...

of ''X''.
*A group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

to the automorphism group of a set ''X'' amounts to a group action
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

on ''X'': indeed, each left ''G''-action on a set ''X'' determines $G\; \backslash to\; \backslash operatorname(X),\; \backslash ,\; g\; \backslash mapsto\; \backslash sigma\_g,\; \backslash ,\; \backslash sigma\_g(x)\; =\; g\; \backslash cdot\; x$, and, conversely, each homomorphism $\backslash varphi:\; G\; \backslash to\; \backslash operatorname(X)$ defines an action by $g\; \backslash cdot\; x\; =\; \backslash varphi(g)x$.
*Let $A,\; B$ be two finite sets of the same cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and $\backslash operatorname(A,\; B)$ the set of all bijection
In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

s $A\; \backslash mathrel\; B$. Then $\backslash operatorname(B)$, which is a symmetric group (see above), acts on $\backslash operatorname(A,\; B)$ from the left freely and transitively; that is to say, $\backslash operatorname(A,\; B)$ is a torsor
In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a gr ...

for $\backslash operatorname(B)$ (cf. #In category theory).
*The automorphism group $G$ of a finite cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

of order
Order or ORDER or Orders may refer to:
* Orderliness
Orderliness is associated with other qualities such as cleanliness
Cleanliness is both the abstract state of being clean and free from germs, dirt, trash, or waste, and the habit of achieving a ...

''n'' is isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to $(\backslash mathbb/n\backslash mathbb)^*$ with the isomorphism given by $\backslash overline\; \backslash mapsto\; \backslash sigma\_a\; \backslash in\; G,\; \backslash ,\; \backslash sigma\_a(x)\; =\; x^a$. In particular, $G$ is an abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

.
*The automorphism group of a field extension
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$L/K$ 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the field extension.
*The automorphism group of the projective ''n''-space over a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

''k'' is the projective linear group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

$\backslash operatorname\_n(k).$
*The automorphism group of a finite-dimensional real Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

$\backslash mathfrak$ has the structure of a (real) Lie group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

(in fact, it is even a linear algebraic group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

: see below). If ''G'' is a Lie group with Lie algebra $\backslash mathfrak$, then the automorphism group of ''G'' has a structure of a Lie group induced from that on the automorphism group of $\backslash mathfrak$.
*Let ''P'' be a finitely generated projective module
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

over a ring ''R''. Then there is an embedding
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

$\backslash operatorname(P)\; \backslash hookrightarrow\; \backslash operatorname\_n(R)$, unique up to inner automorphism
In abstract algebra an inner automorphism is an automorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

s.
In category theory

Automorphism groups appear very naturally incategory theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

.
If ''X'' is an object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Entity, something that is tangible and within the grasp of the senses
** Object (abstract), an object which does not exist at any particular time or pl ...

in a category, then the automorphism group of ''X'' is the group consisting of all the invertible morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s from ''X'' to itself. It is the unit group
In the branch of abstract algebra known as ring theory, a unit of a ring (mathematics), ring R is any element u \in R that has a multiplicative inverse in R: an element v \in R such that
:vu = uv = 1,
where is the multiplicative identity. The s ...

of the endomorphism monoid of ''X''. (For some examples, see PROP.)
If $A,\; B$ are objects in some category, then the set $\backslash operatorname(A,\; B)$ of all $A\; \backslash mathrel\; B$ is a left $\backslash operatorname(B)$-torsor
In mathematics, a principal homogeneous space, or torsor, for a group (mathematics), group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a gr ...

. In practical terms, this says that a different choice of a base point of $\backslash operatorname(A,\; B)$ differs unambiguously by an element of $\backslash operatorname(B)$, 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 $F:\; C\_1\; \backslash to\; C\_2$ is a functor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

mapping $X\_1$ to $X\_2$, then $F$ induces a group homomorphism $\backslash operatorname(X\_1)\; \backslash to\; \backslash operatorname(X\_2)$, as it maps invertible morphisms to invertible morphisms.
In particular, if ''G'' is a group viewed as a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

with a single object * or, more generally, if ''G'' is a groupoid, then each functor $G\; \backslash to\; C$, ''C'' a category, is called an action or a representation of ''G'' on the object $F(*)$, or the objects $F(\backslash operatorname(G))$. Those objects are then said to be $G$-objects (as they are acted by $G$); cf. $\backslash mathbb$-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-dimensionalalgebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

over ''k''). It can be, for example, an associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

or a Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
Now, consider ''k''-linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s $M\; \backslash to\; M$ that preserve the algebraic structure: they form a vector subspace
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$\backslash operatorname\_(M)$ of $\backslash operatorname(M)$. The unit group of $\backslash operatorname\_(M)$ is the automorphism group $\backslash operatorname(M)$. When a basis on ''M'' is chosen, $\backslash operatorname(M)$ is the space of square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and $\backslash operatorname\_(M)$ is the zero set of some polynomial equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, and the invertibility is again described by polynomials. Hence, $\backslash operatorname(M)$ is a linear algebraic group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

over ''k''.
Now base extensions applied to the above discussion determines a functor: namely, for each commutative ring
In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative.
Definition and first e ...

''R'' over ''k'', consider the ''R''-linear maps $M\; \backslash otimes\; R\; \backslash to\; M\; \backslash otimes\; R$ preserving the algebraic structure: denote it by $\backslash operatorname\_(M\; \backslash otimes\; R)$. Then the unit group of the matrix ring $\backslash operatorname\_(M\; \backslash otimes\; R)$ over ''R'' is the automorphism group $\backslash operatorname(M\; \backslash otimes\; R)$ and $R\; \backslash mapsto\; \backslash operatorname(M\; \backslash otimes\; R)$ is a group functor: a functor from the category of commutative rings
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

over ''k'' to the category of groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. 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 $\backslash operatorname(M)$.
In general, however, an automorphism group functor may not be represented by a scheme.
See also

*Outer automorphism groupIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Level structure, a trick to kill an automorphism group
*Holonomy group
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...

References

* * * * * {{cite book , author-link=William C. Waterhouse , first=William C. , last=Waterhouse , title=Introduction to Affine Group Schemes , publisher=Springer Verlag , series=Graduate Texts in Mathematics , volume=66 , orig-year=1979 , year=2012 , isbn=9781461262176 , url=https://books.google.com/books?id=SpfwBwAAQBAJExternal links

*https://mathoverflow.net/questions/55042/automorphism-group-of-a-scheme Group automorphisms