More generally, for an object in some
category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism is an automorphism if there is a morphism such that where is the
identity morphism of X. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the
identity function, and is often called the trivial automorphism
The automorphisms of an object X form a
group under
composition of
morphisms, which is called the automorphism group of X. This results straightforwardly from the definition of a category.
The automorphism group of an object X in a category C is often denoted AutC(X), or simply Aut(X) if the category is clear from context.
Examples
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, 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
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.[1]
In
linear algebra, an endomorphism of a
vector spaceV is a
linear operatorV → 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).)
The field of the
rational numbers has no other automorphism than the identity, since an automorphism must fix the
additive identity0 and the
multiplicative identity1; the sum of a finite number of 1 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 of the
real numbers has no other automorphism than the identity. Indeed, the rational numbers must be fixed by every automorphism, per above; an automorphism must preserve inequalities since is equivalent to 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 automorphism group of the
quaternions (H) as a ring are the inner automorphisms, by the
Skolem–Noether theorem: maps of the form a ↦ bab−1.[4] This group is
isomorphic to
SO(3), the group of rotations in 3-dimensional space.
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:
An automorphism of a differentiable
manifoldM 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
One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician
William Rowan Hamilton in 1856, in his
icosian calculus, where he discovered an order two automorphism,[5] writing:
so that is a new fifth root of unity, connected with the former fifth root by relations of perfect reciprocity.
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 φa : G → G given by φa(g) = aga−1 (or a−1ga; 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 groupAut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the
cosets that contain the outer automorphisms.