Automorphism
Introduction
In the field of mathematics, an automorphism is a morphism from a mathematical object to itself that is both an isomorphism and a bijection. Automorphisms are a central concept in various areas of mathematics, including group theory, ring theory, field theory, and topology. They are used to study the structure of objects by examining their symmetries and self-mappings. The set of all automorphisms of an object forms a group under the operation of composition, known as the automorphism group.
Automorphisms in Group Theory
In group theory, an automorphism of a group \( G \) is a bijective group homomorphism from \( G \) to itself. The set of all automorphisms of a group \( G \), denoted by \( \text{Aut}(G) \), forms a group under the operation of composition. This group is called the automorphism group of \( G \).
Inner and Outer Automorphisms
Automorphisms can be classified into inner and outer automorphisms. An inner automorphism is an automorphism that can be expressed as conjugation by an element of the group. Specifically, for any element \( g \in G \), the map \( \phi_g: G \to G \) defined by \( \phi_g(x) = gxg^{-1} \) for all \( x \in G \) is an inner automorphism. The set of all inner automorphisms forms a normal subgroup of \( \text{Aut}(G) \), denoted by \( \text{Inn}(G) \).
An outer automorphism is an automorphism that is not inner. The quotient group \( \text{Out}(G) = \text{Aut}(G) / \text{Inn}(G) \) is called the outer automorphism group of \( G \). Outer automorphisms provide insight into the structure of the group that is not apparent from the inner automorphisms alone.
Examples
1. **Cyclic Groups**: Every automorphism of a cyclic group is determined by its action on a generator. For a cyclic group \( \mathbb{Z}_n \), the automorphism group is isomorphic to the group of units \( \mathbb{Z}_n^* \).
2. **Symmetric Groups**: The symmetric group \( S_n \) has a rich structure of automorphisms. For \( n \neq 6 \), the automorphism group of \( S_n \) is isomorphic to \( S_n \) itself. However, \( S_6 \) has an exceptional outer automorphism.
3. **Abelian Groups**: For a finite abelian group, the automorphism group can be quite complex. For instance, the automorphism group of \( \mathbb{Z}_p^n \), where \( p \) is a prime, is isomorphic to the general linear group \( \text{GL}(n, \mathbb{Z}_p) \).
Automorphisms in Ring Theory
In ring theory, an automorphism of a ring \( R \) is a bijective ring homomorphism from \( R \) to itself. The set of all automorphisms of a ring forms a group under composition, similar to the case in group theory.
Structure and Examples
1. **Field Automorphisms**: A significant case of ring automorphisms is when the ring is a field. The automorphisms of a field \( F \) that fix a subfield \( K \) form a group known as the Galois group, which plays a crucial role in Galois theory.
2. **Matrix Rings**: Consider the ring of \( n \times n \) matrices over a field \( F \), denoted by \( M_n(F) \). The automorphisms of this ring are closely related to the invertible matrices, forming a group isomorphic to \( \text{GL}(n, F) \).
3. **Polynomial Rings**: The automorphisms of polynomial rings can be quite intricate. For example, the automorphism group of the polynomial ring \( F[x] \) over a field \( F \) is trivial, consisting only of the identity map.
Automorphisms in Field Theory
In field theory, automorphisms are particularly important in the study of field extensions. An automorphism of a field \( F \) is a bijective map from \( F \) to itself that preserves the field operations.
Galois Theory
Galois theory provides a profound connection between field extensions and group theory through the concept of automorphisms. The Galois group of a field extension \( E/F \) is the group of field automorphisms of \( E \) that fix \( F \). This group encodes significant information about the structure of the field extension.
1. **Normal Extensions**: A field extension \( E/F \) is normal if every irreducible polynomial in \( F[x] \) that has a root in \( E \) splits completely in \( E \). The Galois group of a normal extension is particularly well-behaved.
2. **Separable Extensions**: An extension is separable if every element of the extension is the root of a separable polynomial over the base field. The Galois group of a separable extension is finite.
3. **Fundamental Theorem of Galois Theory**: This theorem establishes a correspondence between the subgroups of the Galois group and the intermediate fields of the extension. It provides a powerful tool for solving polynomial equations and understanding the symmetries of algebraic structures.
Automorphisms in Topology
In topology, automorphisms are often referred to as homeomorphisms when considering topological spaces. A homeomorphism is a bijective continuous function with a continuous inverse, mapping a topological space onto itself.
Homeomorphisms and Topological Groups
1. **Topological Groups**: A topological group is a group equipped with a topology such that the group operations are continuous. The automorphisms of a topological group are homeomorphisms that are also group homomorphisms.
2. **Manifolds**: In the context of manifolds, automorphisms are diffeomorphisms, which are smooth bijections with smooth inverses. The study of diffeomorphisms is central to differential topology and geometry.
3. **Fixed Point Theorems**: Automorphisms in topology often lead to interesting results such as fixed point theorems. For instance, the Brouwer Fixed Point Theorem states that any continuous function from a closed disk to itself has at least one fixed point.
Automorphisms in Algebraic Geometry
In algebraic geometry, automorphisms are morphisms of algebraic varieties that are bijective and have bijective inverses. These automorphisms preserve the algebraic structure and are crucial in understanding the symmetries of varieties.
Algebraic Varieties and Schemes
1. **Projective Varieties**: Automorphisms of projective varieties are often studied in the context of projective transformations. These transformations preserve the incidence relations and the dimension of the varieties.
2. **Schemes**: In the language of schemes, automorphisms are morphisms that are isomorphisms. The study of automorphisms of schemes is a deep area of research, involving the interplay between algebraic and geometric properties.
3. **Moduli Spaces**: Automorphisms play a key role in the theory of moduli spaces, which parametrize families of algebraic varieties. Understanding the automorphisms of these spaces helps in classifying varieties and understanding their deformations.