• In
set theory, an arbitrary
permutation of the elements of a set
X is an automorphism. The automorphism group of
X is also called the symmetric group on
X. • In
elementary arithmetic, the set of
integers, , 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
image is the group Inn(
G) of
inner automorphisms and whose
kernel is the
center of
G. Thus, if
G has
trivial center it can be embedded into its own automorphism group. • 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). (The algebraic structure of
all endomorphisms of V is itself an algebra over the same base field as
V, whose
invertible elements precisely consist of GL(
V).) • A field automorphism is a
bijective ring homomorphism from a
field to itself. • The field \Q of the
rational numbers has no other automorphism than the identity, since an automorphism must fix the
additive identity and the
multiplicative identity ; the sum of a finite number of must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes all
integers); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism. • The field \R of the
real numbers has no automorphisms other than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since x is equivalent to \exists z\mid y-x=z^2, and the latter property is preserved by every automorphism; finally every real number must be fixed since it is the
least upper bound of a sequence of rational numbers. • The field \Complex of the
complex numbers has a unique nontrivial automorphism that fixes the real numbers. It is the
complex conjugation, which maps i to -i. The
axiom of choice implies the existence of
uncountably many automorphisms that do not fix the real numbers. • The study of automorphisms of
algebraic field extensions is the starting point and the main object of
Galois theory. • The automorphism group of the
quaternions () as a ring are the inner automorphisms, by the
Skolem–Noether theorem: maps of the form . This group is
isomorphic to
SO(3), the group of rotations in 3-dimensional space. • The automorphism group of the
octonions () is the
exceptional Lie group G2. • 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. • In
geometry, an automorphism may be called a
motion of the space. Specialized terminology is also used: • In
metric 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
biholomorphic map (also called a
conformal map), from a surface to itself. For example, the automorphisms of the
Riemann sphere are
Möbius transformations. • An automorphism of a differentiable
manifold M is a
diffeomorphism from
M to itself. The automorphism group is sometimes denoted Diff(
M). • In
topology, morphisms between topological spaces are called
continuous maps, and an automorphism of a topological space is a
homeomorphism of the space to itself, or self-homeomorphism (see
homeomorphism group). In this example it is
not sufficient for a morphism to be bijective to be an isomorphism. ==History==