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{{Short description|Isomorphism of an object to itself}}File:Klein-automorphism.svg|thumb|right|400px|An w:Automorphism|automorphism]] of the w:Klein_four-group|Klein four-group]] shown as a mapping between two w:Cayley_graph|Cayley graphs]], a permutation in w:Cycle_notation|cycle notation]], and a mapping between two w:Cayley_table|Cayley tables]].]]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: Xto X is an automorphism if there is a morphism g: Xto X such that gcirc f= fcirc g = operatorname {id}_X, where operatorname {id}_X is the identity morphism of {{mvar|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 {{mvar|X}} form a group under composition of morphisms, which is called the automorphism group of {{mvar|X}}. This results straightforwardly from the definition of a category.The automorphism group of an object {{math|X}} in a category {{math|C}} is often denoted {{math|AutC(X)}}{{math|}}, 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.BOOK,weblink 376, §7.5.5 Automorphisms, Mathematical foundations of computational engineering, Felix Pahl translation, PJ Pahl, R Damrath, 3-540-67995-2, 2001, Springer,
• 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 {{math|0}} and the multiplicative identity {{math|1}}; the sum of a finite number of {{math|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 R 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 x