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Automorphism

In mathematics, an automorphism is an isomorphism of a mathematical structure onto itself, derived from the Greek words auto ("self") and morphosis ("to form" or "shape"), representing a symmetry that preserves all the structure's properties. The collection of all such automorphisms for a given structure forms a group under function composition, called the automorphism group, which encodes the structure's symmetries and is fundamental in studying its properties. Automorphisms arise across various branches of mathematics, particularly in and . In group theory, an automorphism of a group G is a bijective from G to itself; for the \mathbb{Z}_n of order n, the \operatorname{Aut}(\mathbb{Z}_n) is isomorphic to the of units n, U(n), consisting of integers coprime to n. For the V_4, \operatorname{Aut}(V_4) has six elements and is isomorphic to the S_3. In ring theory and field theory, an automorphism is a bijective from the structure to itself that preserves addition and multiplication; for example, the field of rational numbers \mathbb{Q} has only the trivial automorphism (the map). In , an automorphism of a graph H = (V, E) is a bijective map \phi: V \to V such that vertices x and y are adjacent \phi(x) and \phi(y) are adjacent, preserving the graph's adjacency relations. The automorphism group of the on four vertices is the S_4, reflecting all possible relabelings of its vertices. Automorphisms also appear in , such as conformal self-maps of the , and their study reveals deep connections between algebraic structures and their invariant properties.

Definition and Basics

Formal Definition

In abstract algebra and related fields, the foundational concepts leading to automorphisms are homomorphisms and isomorphisms. A is a map between two mathematical structures of the same type that preserves the operations and relations defined on those structures. An is a bijective homomorphism, meaning it is both injective and surjective, thereby establishing a one-to-one correspondence that maintains the structural properties. An is an from a to itself, representing a of the object that preserves all its intrinsic operations and relations. More formally, for a S equipped with operations or relations (such as a , or ), an automorphism \phi: S \to S is a bijective satisfying the preservation conditions specific to the structure. For instance, in a multiplicative structure, \phi(ab) = \phi(a)\phi(b) for all a, b \in S; analogous conditions hold for additive structures, where \phi(a + b) = \phi(a) + \phi(b), or relational ones, where \phi maps relations to corresponding relations. The collection of all automorphisms of a structure S, under the operation of , forms a group known as the , denoted \operatorname{Aut}(S). This group arises because the is an automorphism, of automorphisms yields another automorphism, and every automorphism has an inverse that is also an automorphism.

Relation to Isomorphisms

An automorphism of a is a special case of an where the mapping is from the structure to itself, preserving all relevant operations and relations in a bijective manner. This self-mapping ensures that the 's properties are maintained bidirectionally, distinguishing it from general isomorphisms between distinct structures. In contrast to endomorphisms, which are structure-preserving mappings from a structure to itself but not necessarily bijective, automorphisms are precisely the invertible endomorphisms. The invertibility requirement guarantees that the mapping can be reversed while still preserving the structure, making automorphisms the bijective subset of endomorphisms. If \phi is an automorphism of a structure, then its inverse \phi^{-1} is also an automorphism, as the inverse of an isomorphism is itself an isomorphism, and applying it to the same structure maintains bijectivity and preservation properties. This closure under inversion is a fundamental aspect that allows the collection of automorphisms to form a group under composition. Automorphisms fundamentally capture the symmetries inherent in a , representing all ways to transform the object while leaving its essential properties unchanged. These symmetries provide insight into the object's intrinsic , often revealing equivalences that simplify or .

Automorphism Groups

Structure and Properties

The \Aut(S) of an S consists of all automorphisms of S, with the group operation defined by of functions. This set forms a group because the identity mapping on S serves as the , as it preserves all operations and relations of S. is closed: if \phi and \psi are automorphisms, then \phi \circ \psi is also an automorphism, since it bijectively maps S to itself while preserving structure. Additionally, every automorphism \phi has an inverse \phi^{-1}, which is itself an automorphism, ensuring the group axioms are satisfied. \Aut(S) embeds as a of the \Sym(S) on the underlying set of S, where the comprises all of S under . This arises because every automorphism is a , and the test confirms the inclusion: \Aut(S) is nonempty, closed under the operation, and closed under inverses. For a finite S with |S| = n, implies that the order of \Aut(S) divides n!, the order of \Sym(S). In the case, \Aut(S) may be , exhibiting potentially more intricate topological or properties not present in finite scenarios. Key properties of \Aut(S) include its center Z(\Aut(S)), the subgroup of automorphisms that commute with every element of \Aut(S) under composition, which captures "central symmetries" of the structure. Normal subgroups of \Aut(S) play a central role in its quotient structure; notably, the inner automorphism subgroup forms a normal subgroup. These properties distinguish \Aut(S) as a rich algebraic object, with finite cases often yielding computable structures while infinite cases require advanced tools from or .

Inner Automorphism Subgroup

In , an of a group G is an automorphism induced by conjugation by an element of G. Specifically, for each g \in G, the map \phi_g: G \to G defined by \phi_g(x) = g x g^{-1} for all x \in G is an automorphism of G. This follows from the properties of conjugation preserving the group operation: \phi_g(x y) = g (x y) g^{-1} = (g x g^{-1}) (g y g^{-1}) = \phi_g(x) \phi_g(y). The collection of all inner automorphisms forms a of the \Aut(G), denoted \Inn(G) = \{ \phi_g \mid g \in G \}. The map \psi: G \to \Aut(G) given by \psi(g) = \phi_g is a , as composition satisfies \phi_g \circ \phi_h = \phi_{g h} for all g, h \in G. The of \psi is precisely Z(G) = \{ z \in G \mid z x = x z \ \forall x \in G \}, since \phi_z = \mathrm{id}_G if and only if z commutes with every element of G. By the first isomorphism theorem, this yields \Inn(G) \cong G / Z(G). The subgroup \Inn(G) is in \Aut(G). To see this, consider any \alpha \in \Aut(G) and \phi_g \in \Inn(G); then \alpha \circ \phi_g \circ \alpha^{-1} = \phi_{\alpha(g)}, which is again an inner automorphism, confirming that \Inn(G) is invariant under conjugation by elements of \Aut(G). This highlights the central role of inner automorphisms within the full .

Examples in Structures

In Groups

In group theory, automorphisms preserve the group operation and structure, and concrete examples illustrate their role in various group classes. For cyclic groups, consider the finite cyclic group \mathbb{Z}_n of order n, generated by 1 under addition modulo n. Any automorphism \phi is determined by \phi(1) = k, where k is coprime to n to ensure \phi is bijective, as the order of k must be n. Thus, the automorphism group \operatorname{Aut}(\mathbb{Z}_n) is isomorphic to (\mathbb{Z}/n\mathbb{Z})^\times, the multiplicative group of units modulo n, which consists of integers from 1 to n-1 coprime to n. For n = p a prime, (\mathbb{Z}/p\mathbb{Z})^\times has order p-1 and is cyclic, generated by a primitive root modulo p. Symmetric groups provide another key example. The S_n on n letters, consisting of all permutations of \{1, 2, \dots, n\}, has \operatorname{Aut}(S_n) isomorphic to S_n itself for n \neq 6, meaning all automorphisms are inner, arising from conjugation by elements of S_n. In this case, inner automorphisms via conjugation permute the transpositions while preserving the group's structure. However, for n=6, \operatorname{Aut}(S_6) is larger than S_6, admitting outer automorphisms that do not arise from conjugation. For abelian groups, particularly free abelian groups, automorphisms correspond to linear transformations. The free abelian group \mathbb{Z}^k of rank k, additively generated by the standard basis \{e_1, \dots, e_k\}, has \operatorname{Aut}(\mathbb{Z}^k) isomorphic to the general linear group \mathrm{GL}(k, \mathbb{Z}), the group of k \times k invertible matrices over \mathbb{Z} with determinant \pm 1. Each such matrix defines an automorphism by sending the basis to new generators that form a \mathbb{Z}-basis for \mathbb{Z}^k. A specific finite abelian example is the Klein four-group V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 = \{0, a, b, a+b\} under addition modulo 2, where the three non-identity elements have order 2. Here, \operatorname{Aut}(V_4) \cong S_3, as automorphisms permute these three elements arbitrarily while fixing the identity, reflecting the symmetric action on indistinguishable order-2 elements.

In Rings and Fields

An automorphism of a ring R is a bijective ring homomorphism \phi: R \to R that preserves both addition and multiplication, as well as the multiplicative identity $1_R. These maps are more constrained than group automorphisms because they must respect two operations simultaneously. For example, in the ring of integers \mathbb{Z}, any automorphism must send the generator 1 to itself, and thus fix all multiples of 1, making \operatorname{Aut}(\mathbb{Z}) trivial and consisting solely of the identity. In field theory, automorphisms similarly preserve addition and multiplication but are defined relative to a base field. For a field extension K/F, a field automorphism of K fixing F pointwise must fix the prime subfield of F elementwise, as it is generated by the multiplicative identity. When K/F is a Galois extension, the Galois group \operatorname{Gal}(K/F) coincides exactly with the automorphism group \operatorname{Aut}(K/F). Specific examples illustrate the restrictiveness of field automorphisms. The automorphism group \operatorname{Aut}(\mathbb{Q}) of the rational numbers is trivial, as any automorphism fixes \mathbb{Z} (generated by 1) and thus all quotients. Similarly, \operatorname{Aut}(\mathbb{R}) is trivial; any automorphism preserves squares (hence positivity) and the order structure, fixing \mathbb{Q} densely and extending uniquely to all reals. In contrast, the only continuous field automorphism of \mathbb{C} (over \mathbb{Q}) is complex conjugation (besides the identity), though the full automorphism group is vastly larger, of cardinality $2^{2^{\aleph_0}}. Complex conjugation is given by \overline{z} = a - bi for z = a + bi. For rings over a k, the \operatorname{Aut}_k(k) consists of the affine transformations x \mapsto ax + b with a \in k^\times and b \in k, forming the affine group \operatorname{Aff}_1(k) \cong k^\times \ltimes k. This structure arises because any automorphism is uniquely determined by the image of x, which must be a linear to preserve the ring's grading and irreducibility properties.

Advanced Concepts

Outer Automorphisms

The outer automorphism group of a S, denoted \mathrm{Out}(S), is defined as the \mathrm{Out}(S) = \mathrm{Aut}(S) / \mathrm{Inn}(S), where \mathrm{Aut}(S) is the full of S and \mathrm{Inn}(S) is its consisting of s. The elements of \mathrm{Out}(S) are the cosets of \mathrm{Inn}(S) in \mathrm{Aut}(S), each corresponding to an of automorphisms that agree up to composition with an inner automorphism. This construction captures the automorphisms of S modulo those induced by conjugation by elements of S itself. The group \mathrm{Out}(S) quantifies the "non-conjugacy" symmetries of S, representing those automorphisms that cannot be obtained via internal conjugation and thus reflect external or exceptional symmetries of the . It may be trivial—for instance, when every automorphism is inner—or non-trivial, and its can range from abelian to non-abelian, depending on S. For many groups, \mathrm{Out}(S) is finite and solvable, highlighting the limited extent of such external symmetries in algebraic . A prominent example occurs with the S_6, where \mathrm{Out}(S_6) \cong \mathbb{Z}_2. This arises because S_6 admits exactly one outer automorphism up to with inner automorphisms, yielding |\mathrm{Aut}(S_6)| = 2 \cdot |S_6| = [1440](/page/1440), in contrast to other s S_n (for n \neq 6) which have trivial outer automorphism groups. The exceptional nature of S_6 stems from its duality with the \mathrm{Sp}(4,2), enabling this unique non-inner . For finite groups G, the outer automorphism group \mathrm{Out}(G) relates to the M(G) in the study of covering groups, particularly for perfect groups where the automorphism group of the Schur cover \tilde{G} involves extensions incorporating both \mathrm{Out}(G) and actions on M(G). Computations for finite simple groups of Lie type illustrate this interplay, as the structures of \mathrm{Out}(G) and M(G) together determine the full automorphism tower and covering extensions.

Applications in Geometry and Graphs

In geometry, automorphisms preserve the underlying structure of spaces, with isometries of forming a key example; these include rotations, reflections, translations, and glide reflections that maintain distances and orientations. The of the , viewed as an , is the affine group, comprising all invertible affine transformations that map the space to itself bijectively while preserving and ratios of distances along lines. Crystallographic groups extend these ideas to discrete symmetries in and physics, acting as subgroups of the of that preserve a lattice structure, such as atomic arrangements in . In three dimensions, there are 230 such groups, obtained by combining the 14 Bravais lattices with the 32 crystallographic point groups, which classify the possible symmetry operations compatible with translational periodicity and underpin phenomena like diffraction patterns in . In , automorphisms are bijections on the set that preserve adjacency, effectively capturing the symmetries of the graph's structure. For the K_n on n vertices, where every pair of vertices is adjacent, the \mathrm{Aut}(K_n) is isomorphic to the S_n, as any of vertices maintains the full connectivity. A notable example is the , a 3-regular graph with 10 vertices and 15 edges, whose automorphism group has order 120 and is isomorphic to S_5, reflecting its high degree of symmetry despite being non-Hamiltonian. Outer automorphisms of graph automorphism groups can model exceptional symmetries in combinatorial structures, such as those arising from exotic embeddings that extend standard permutation actions beyond inner conjugations.

Historical Development

Origins in Geometry

The concept of automorphisms originated in the study of geometric symmetries, where self-mappings of figures preserve distances, angles, and incidence relations. In ancient Greek mathematics, particularly in Euclid's Elements (circa 300 BCE), congruences were established through the method of superposition, which implicitly treats rotations and reflections as rigid transformations mapping one figure onto another while maintaining their properties. For instance, Proposition I.4 demonstrates that if two triangles have two sides and the included angle equal, then the triangles are congruent by superposing one upon the other, aligning corresponding parts so that vertices coincide, thereby proving equality of the bases and remaining angles. This approach, rooted in practical constructions, prefigures automorphisms as structure-preserving maps, with rotations around a point and reflections over a line serving as basic examples of such symmetries in plane geometry. During the 18th and early 19th centuries, geometric investigations increasingly emphasized transformations that preserve specific properties, laying groundwork for group-theoretic interpretations without yet formalizing abstract groups. Mathematicians like and advanced , focusing on collineations—transformations preserving incidence between points and lines—as invariant under perspective projections. , in 1827, classified geometric configurations by properties invariant under certain transformation classes, such as similarities and affinities, effectively describing early notions of symmetry groups acting on space. Jakob Steiner's in 1832 further highlighted these transformations in studying conic sections and polygons, treating them as operations that reorder elements while conserving relational structures. A pivotal advancement came with Felix Klein's 1872 Erlangen program, which systematically classified geometries according to their associated groups of automorphisms—transformations preserving the fundamental incidence relations of the space. In his inaugural address at the University of Erlangen, Klein proposed viewing a geometry as the study of under a "principal group" of collineations for , or metric-preserving motions for , thereby unifying diverse branches like affine and hyperbolic geometries under group actions. For example, is characterized by the group of rigid motions (rotations, translations, reflections) that leave distances , while relies on broader collineations preserving only point-line incidences. Klein's framework emphasized automorphisms as the "Hauptgruppe" defining geometric properties through their invariance, marking a transition from transformations to organized studies. These geometric developments predate the abstract theory of groups, with early actions manifesting as collections of transformations composing under and inversion, applied to figures like polygons and curves to reveal symmetries. By the mid-19th century, such actions on geometric objects, as in the of regular polyhedra via rotational symmetries, anticipated modern algebraic generalizations of automorphisms.

Developments in Abstract Algebra

In the early , the study of automorphisms gained prominence in group theory through the foundational work of William Burnside. His 1897 book Theory of Groups of Finite Order, revised in 1911, systematically explored the structure of finite groups, including the role of automorphisms in determining isomorphisms and the holomorph of a group as the of the group with its . This framework highlighted automorphisms as essential for classifying groups and understanding their symmetries, influencing subsequent developments in . A key advancement came from in 1911, who identified the first known non-trivial outer automorphism in the S_6, demonstrating that the of S_6 is larger than its group by a factor of 2. Schur's discovery, detailed in his analysis of group representations, revealed an exceptional case where conjugacy classes are not preserved under all automorphisms, challenging the expectation that symmetric groups beyond S_2 are complete. This work extended the understanding of outer automorphisms beyond inner ones, paving the way for deeper investigations into group structures in the 1920s. The integration of automorphisms into , originating with Évariste Galois's ideas in the 1830s, was formalized in the , emphasizing the as the of the . Emil Artin's 1944 lectures, published as , presented the subject through field automorphisms and their fixed fields, establishing the fundamental theorem as a between subfields and subgroups without relying on solvability first. This approach solidified automorphisms as central to solvability criteria, bridging field theory with group automorphisms and influencing ring and field extensions. Post-World War II, automorphisms played a crucial role in , particularly in modular representations and . Richard Brauer's extensions of Frobenius's work in the 1940s and 1950s on , including the development of block theory and decomposition numbers, aided the , where group automorphisms are essential. By the 1960s, the exceptional outer automorphism of S_6, discovered by Schur, continued to be analyzed in the context of , confirming its impact on irreducible characters and contributing to the ongoing classification efforts. From the onward, category-theoretic perspectives extended the algebraic view of automorphisms, treating them as endomorphisms in categories of algebraic structures. Alexander Grothendieck's applications in , such as topos theory and schemes, framed automorphism groups abstractly, limited to algebraic contexts like sheaves and motives. This shift emphasized universal properties over explicit computations, influencing modern while remaining grounded in group, , and theories.