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Automorphism

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From Wikipedia, the free encyclopedia

Isomorphism of an object to itself

Anautomorphism of theKlein four-group shown as a mapping between twoCayley graphs, a permutation incycle notation, and a mapping between twoCayley tables.

Inmathematics, anautomorphism is anisomorphism from amathematical object to itself. It is, in some sense, asymmetry of the object, and a way ofmapping the object to itself while preserving all of its structure. Theset of all automorphisms of an object forms agroup, called theautomorphism group. It is, loosely speaking, thesymmetry group of the object.

Definition

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In analgebraic structure such as agroup, aring, orvector space, anautomorphism is simply abijective 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, andlinear operator.)

More generally, for an object in somecategory, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism $f:X\to X$ is an automorphism if there is a morphism $g:X\to X$ such that $g\circ f=f\circ g=\operatorname {id} _{X},$ where $\operatorname {id} _{X}$ is theidentity morphism ofX. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply theidentity function, and is often called thetrivial automorphism.

Automorphism group

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Main article:Automorphism group

The automorphisms of an objectX form agroup undercomposition ofmorphisms, which is called theautomorphism group ofX. This results straightforwardly from the definition of a category.

The automorphism group of an objectX in a categoryC is often denotedAut_C(X), or simply Aut(X) if the category is clear from context.

Examples

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Inset theory, an arbitrarypermutation of the elements of a setX is an automorphism. The automorphism group ofX is also called the symmetric group onX.
Inelementary arithmetic, the set ofintegers,⁠ $\mathbb {Z}$ ⁠, 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 anyabelian group, but not of a ring or field.
A group automorphism is agroup isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every groupG there is a natural group homomorphismG → Aut(G) whoseimage is the group Inn(G) ofinner automorphisms and whosekernel is thecenter ofG. Thus, ifG hastrivial center it can be embedded into its own automorphism group.^[1]
Inlinear algebra, anendomorphism of avector spaceV is alinear operatorV →V. An automorphism is an invertible linear operator onV. When the vector space is finite-dimensional, the automorphism group ofV is the same as thegeneral linear group, GL(V). (The algebraic structure ofall endomorphisms ofV is itself an algebra over the same base field asV, whoseinvertible elements precisely consist of GL(V).)
A field automorphism is abijectivering homomorphism from afield to itself.
- The field $\mathbb {Q}$ of therational numbers has no other automorphism than the identity, since an automorphism must fix theadditive identity0 and themultiplicative identity1; the sum of a finite number of1 must be fixed, as well as the additive inverses of these sums (that is, the automorphism fixes allintegers); finally, since every rational number is the quotient of two integers, all rational numbers must be fixed by any automorphism.
- The field $\mathbb {R}$ of thereal 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 $x<y$ 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 theleast upper bound of a sequence of rational numbers.
- The field $\mathbb {C}$ of thecomplex numbers has a unique nontrivial automorphism that fixes the real numbers. It is thecomplex conjugation, which maps $i {\displaystyle i}$ to $-i.$ Theaxiom of choice implies the existence ofuncountably many automorphisms that do not fix the real numbers.^[2]^[3]
- The study of automorphisms ofalgebraic field extensions is the starting point and the main object ofGalois theory.
The automorphism group of thequaternions (⁠ $\mathbb {H}$ ⁠) as a ring are the inner automorphisms, by theSkolem–Noether theorem: maps of the forma ↦bab⁻¹.^[4] This group isisomorphic toSO(3), the group of rotations in 3-dimensional space.
The automorphism group of theoctonions (⁠ $\mathbb {O}$ ⁠) is theexceptional Lie group G₂.
Ingraph theory anautomorphism 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.
Ingeometry, an automorphism may be called amotion of the space. Specialized terminology is also used:
- Inmetric geometry an automorphism is a self-isometry. The automorphism group is also called theisometry group.
- In the category ofRiemann surfaces, an automorphism is abiholomorphic map (also called aconformal map), from a surface to itself. For example, the automorphisms of theRiemann sphere areMöbius transformations.
- An automorphism of a differentiablemanifoldM is adiffeomorphism fromM to itself. The automorphism group is sometimes denoted Diff(M).
- Intopology, morphisms between topological spaces are calledcontinuous maps, and an automorphism of a topological space is ahomeomorphism of the space to itself, or self-homeomorphism (seehomeomorphism group). In this example it isnot sufficient for a morphism to be bijective to be an isomorphism.

History

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One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematicianWilliam Rowan Hamilton in 1856, in hisicosian calculus, where he discovered an order two automorphism,^[5] writing:

so that $\mu$ is a new fifth root of unity, connected with the former fifth root $\lambda$ by relations of perfect reciprocity.

Inner and outer automorphisms

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Main articles:Inner automorphism andOuter automorphism group

In some categories—notablygroups,rings, andLie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, theinner automorphisms are the conjugations by the elements of the group itself. For each elementa of a groupG, conjugation bya is the operationφ_a :G →G given byφ_a(g) =aga⁻¹ (ora⁻¹ga; usage varies). One can easily check that conjugation bya is a group automorphism. The inner automorphisms form anormal subgroup of Aut(G), denoted by Inn(G); this is calledGoursat's lemma.

The other automorphisms are calledouter automorphisms. Thequotient groupAut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are thecosets that contain the outer automorphisms.

The same definition holds in anyunital ring oralgebra wherea is anyinvertible element. ForLie algebras the definition is slightly different.

References

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^PJ Pahl, R Damrath (2001)."§7.5.5 Automorphisms".Mathematical foundations of computational engineering (Felix Pahl translation ed.). Springer. p. 376.ISBN 3-540-67995-2.
^Yale, Paul B. (May 1966)."Automorphisms of the Complex Numbers"(PDF).Mathematics Magazine.39 (3):135–141.doi:10.2307/2689301.JSTOR 2689301.
^Lounesto, Pertti (2001),Clifford Algebras and Spinors (2nd ed.), Cambridge University Press, pp. 22–23,ISBN 0-521-00551-5
^Handbook of algebra, vol. 3,Elsevier, 2003, p. 453
^Sir William Rowan Hamilton (1856)."Memorandum respecting a new System of Roots of Unity"(PDF).Philosophical Magazine.12: 446.Archived(PDF) from the original on 2022-10-09.