Automorphism

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

In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.)

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

f

:

X

→

X

{\displaystyle f:X\to X}

is an automorphism if there is a morphism

g

:

X

→

X

{\displaystyle g:X\to X}

such that

g

∘

f

=

f

∘

g

=

id

X

,

{\displaystyle g\circ f=f\circ g=\operatorname {id} _{X},}

where

id

X

{\displaystyle \operatorname {id} _{X}}

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.

Automorphism group

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

{\displaystyle \mathbb {Z} }

⁠, considered as a group under addition, has a unique nontrivial automorphism: negation.