TheInfoList

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 the
symmetric 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 $G \to \operatorname\left(X\right), \, g \mapsto \sigma_g, \, \sigma_g\left(x\right) = g \cdot x$, 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 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 $\operatorname\left(A, B\right)$ 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 $A \mathrel B$. 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 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 $\operatorname\left(B\right)$ (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 $\left(\mathbb/n\mathbb\right)^*$ 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 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 ...
$\operatorname_n\left(k\right).$ *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 ...
$\mathfrak$ 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 $\mathfrak$, 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
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 ...
$\operatorname\left(P\right) \hookrightarrow \operatorname_n\left(R\right)$, 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 in
category 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 $\operatorname\left(A, B\right)$ of all $A \mathrel B$ is a left $\operatorname\left(B\right)$-
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 $\operatorname\left(A, B\right)$ 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 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 $\operatorname\left(X_1\right) \to \operatorname\left(X_2\right)$, as it maps invertible morphisms to invertible morphisms. In particular, if ''G'' is a group viewed as a
category 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 $G \to C$, ''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. $\mathbb$-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 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 $M \to M$ 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 ...
$\operatorname_\left(M\right)$ 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 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 $\operatorname_\left(M\right)$ 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, $\operatorname\left(M\right)$ 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 $M \otimes R \to M \otimes R$ 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 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 $\operatorname\left(M\right)$. In general, however, an automorphism group functor may not be represented by a scheme.

*
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=SpfwBwAAQBAJ