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 an object forms a
group, called the 'automorphism group'. It is, loosely speaking, the
symmetry group of the object.
Definition
The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called
category theory. Category theory deals with abstract objects and
morphisms between those objects.
In category theory, an 'automorphism' is an
endomorphism (i.e. a
morphism from an object to itself) which is also an
isomorphism (in the categorical sense of the word).
This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.
In the context of
abstract algebra, for example, a mathematical object is an
algebraic structure such as a
group,
ring, or
vector space. An isomorphism is simply a
bijective homomorphism. (Of course, the definition of a homomorphism depends on the type of algebraic structure; see, for example:
group homomorphism,
ring homomorphism, and
linear operator).
Automorphism group
The automorphisms of an object ''X'' form a
group under composition of
morphisms. This group is called the 'automorphism group' of ''X''. That this is indeed a group is simple to see:
★
Closure: composition of two endomorphisms is another endomorphism.
★
Associativity: composition of functions is ''always'' associative.
★
Identity: the identity is the identity morphism from an object to itself which exists by definition.
★
Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.
The automorphism group of an object ''X'' in a category ''C'' is denoted Aut
''C''(''X''), or simply Aut(''X'') if the category is clear from context.
Examples
★ In
set theory, an automorphism of a set ''X'' is an arbitrary
permutation of the elements of ''X''. The automorphism group of ''X'' is also called the
symmetric group on ''X''.
★ In
elementary arithmetic, the set of
integers, 'Z', considered as a group under addition, has a unique nontrivial automorphism : negation. Considered as a
ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any
abelian group, but not of a ring or field.
★ A group automorphism is a
group isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group ''G'' there is a natural group homomorphism ''G'' → Aut(''G'') whose
kernel is the
center of ''G''. Thus, if ''G'' is centerless it can be embedded into its own automorphism group. (See the discussion on inner automorphisms below).
★ In
linear algebra, an endomorphism of a
vector space ''V'' is a
linear operator ''V'' → ''V''. An automorphism is an invertible linear operator on ''V''. When the vector space is finite-dimensional, the automorphism group of ''V'' is the same as the
general linear group, GL(''V'').
★ A field automorphism is a
bijective ring homomorphism from a
field to itself. In the case of the
rational numbers, 'Q' there are no nontrivial field automorphisms; in the case of the
real numbers 'R' there are no nontrivial
order-preserving field automorphisms (that is, automorphisms of the
ordered field). In the case of the
complex numbers, 'C', there is a unique nontrivial automorphism that sends 'R' into 'R':
complex conjugation, but there are infinitely (
uncountably) many "wild" automorphisms (assuming the
axiom of choice).
[1] Field automorphisms are important to the theory of
field extensions, in particular
Galois extensions. In the case of a Galois extension ''L''/''K'' the
subgroup of all automorphisms of ''L'' fixing ''K'' pointwise is called the
Galois group of the extension.
★ In
graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation.
★ For relations, see .
★
★ In
order theory, see
order automorphism.
★ An automorphism of a differentiable
manifold ''M'' is a
diffeomorphism from ''M'' to itself. The automorphism group is sometimes denoted Diff(''M'').
★ In
Riemannian geometry an automorphism is a self-
isometry. The automorphism group is also called the
isometry group.
★ In the category of
Riemann surfaces, an automorphism is a bijective
biholomorphic map (also called a
conformal map), from a surface to itself. For example, the automorphisms of the
Riemann sphere are
Möbius transformations.
Inner and outer automorphisms
In some categories—notably
groups,
rings, and
Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
In the case of groups, the
inner automorphisms are the conjugations by the elements of the group itself. For each element ''a'' of a group ''G'', conjugation by ''a'' is the operation φ
''a'' : ''G'' → ''G'' given by φ
''a''(''g'') = ''aga''
−1 (or ''a''
−1''ga''; usage varies). One can easily check that conjugation by ''a'' is a group automorphism. The inner automorphisms form a
normal subgroup of Aut(''G''), denoted by Inn(''G''); this is called
Goursat's lemma.
The other automorphisms are called
outer automorphisms. The
quotient group Aut(''G'') / Inn(''G'') is usually denoted by Out(''G''); the non-trivial elements are the cosets that contain the outer automorphisms.
The same definition holds in any
unital ring or
algebra where ''a'' is any
invertible element. For
Lie algebras the definition is slightly different.
See also
★
endomorphism
★
endomorphism ring
★
antiautomorphism
★
Frobenius automorphism
References
1. Automorphisms of the Complex Numbers, , Paul B., Yale, Mathematics Magazine, 1966
External links
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