Definition:Automorphism (Abstract Algebra)

Definition
An automorphism is an isomorphism from an algebraic structure to itself.

This applies to the term isomorphism as used both in the sense of bijective homomorphism as well as that of an order isomorphism.

Hence an automorphism is a permutation which is either a homomorphism or an order isomorphism, depending on context.

Group Automorphism
If $$\left({S, \circ}\right)$$ is a group, then an automorphism $$\phi: \left({S, \circ}\right) \to \left({S, \circ}\right)$$ is called a group automorphism.

Ring Automorphism
If $$\left({R, +, \circ}\right)$$ is a ring, then an automorphism $$\phi: \left({R, +, \circ}\right) \to \left({R, +, \circ}\right)$$ is called a ring automorphism.

Field Automorphism
If $$\left({S, \circ, \ast}\right)$$ is a field, then an automorphism $$\phi: \left({S, \circ, \ast}\right) \to \left({S, \circ, \ast}\right)$$ is called a field automorphism.

R-Algebraic Structure Endomorphism
If $$\left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R$$ is an $R$-algebraic structure, then an automorphism:
 * $$\phi: \left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R \to \left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R$$

is called an $$R$$-Algebraic Structure automorphism.

Group definition

 * : $$\S 7.1$$
 * : Chapter $$\text{II}$$: Problem $$\text{AA}$$
 * : $$\S 8$$: Definition $$8.10$$

Ring definition

 * : $$\S 23$$