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.

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.

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 $$8.10$$