Definition:Group Automorphism
Jump to navigation
Jump to search
Definition
Let $\struct {G, \circ}$ be a group.
Let $\phi: G \to G$ be a (group) isomorphism from $G$ to itself.
Then $\phi$ is a group automorphism.
Examples
Constant Product on Real Numbers
Let $\struct {\R, +}$ denote the real numbers under addition.
Let $\alpha \in \R$ be a real number.
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \alpha x$
Then $f$ is a (group) automorphism if and only if $\alpha \ne 0$.
Also see
- Results about group automorphisms can be found here.
Linguistic Note
The word automorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus automorphism means self structure.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.1$. Homomorphisms
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{AA}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $24$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Definition $8.10$