automorphism
Let an agent explain automorphisms, relay definitions, automorphism groups and examples across algebra, graph theory, topology and dynamics from mathematics sources, describe the named forms, and distinguish automorphisms from endomorphisms, isomorphisms between different objects, homomorphisms and symmetries in physics.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain automorphisms, relay definitions, automorphism groups and examples across algebra, graph theory, topology and dynamics from mathematics sources, describe the named forms, and distinguish automorphisms from endomorphisms, isomorphisms between different objects, homomorphisms and symmetries in physics.
An isomorphism from a mathematical object to itself, preserving its structure, forming a group under composition called the automorphism group, including group automorphisms, with inner automorphisms given by conjugation, graph automorphisms that permute vertices while preserving edges and measure a graph s symmetry, field automorphisms central to Galois theory, deck transformations of covering spaces in topology, and in dynamical systems hyperbolic toral automorphisms such as Arnold s cat map; the registry alias hyperbolic motion names isometries of hyperbolic space, related symmetries. Automorphisms formalise the idea of symmetry across mathematics.
What it is for: Describing symmetries of mathematical structures.
It can be explain the definition; relay automorphism groups; describe named forms; distinguish related maps.
Distinguishing features
Bijective
Structure-preserving
Forms a group
Captures symmetry
What it looks like
Not a physical object; visualised as symmetries of graphs, shapes and structures.
Physical character
Galois theory: 1830s note - field automorphisms
Arnold cat map: 1960s note - hyperbolic toral automorphism
registry parents: endomorphism, isomorphism note
How it is recognised
Structure-preserving self-isomorphism
Inner automorphism, hyperbolic toral automorphism, group automorphism, hyperbolic motion, deck transformation, graph automorphism
Endomorphisms need not be invertible; isomorphisms may map between objects; homomorphisms preserve structure without bijection; physical symmetries act on systems
Related models
is a kind of - in registry terms
forms -
includes -
is contrasted with -
In practice
Families and kinds
group automorphisms
field automorphisms
graph automorphisms
deck transformations
toral automorphisms
Standards and regulation
No regulation
Failure modes and hazards
Confusing endomorphisms and automorphisms
Ignoring the structure being preserved
Loosely filed geometry alias
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What an automorphism is.
Definition.
Definition
Definition.
Definition
Definition.
- What is an automorphism, and how does it differ from endomorphisms, isomorphisms, homomorphisms and physical symmetries? definition
- Is the question about algebra, graphs, topology or dynamics? boundary
Forms
Named forms.
Forms
Forms.
- What are inner, group, graph and hyperbolic toral automorphisms, deck transformations and hyperbolic motions? definition
- Which entry fits the specific form? action
Algebra Algebra.
Science.
Groups
Group automorphisms.
Groups
Groups.
- How do inner and outer automorphisms of groups differ? provenance
- Which references are standard? provenance
Galois
Galois theory.
Galois
Galois.
- How do field automorphisms explain solvability of equations? provenance
- Which sources are cited? provenance
Other fields Other fields.
Science.
Graphs
Graph symmetry.
Graphs
Graphs.
- How are graph automorphism groups computed, and what is the graph isomorphism problem? provenance
- Which entry fits graph isomorphism problem? action
Dynamics
Toral automorphisms.
Dynamics
Dynamics.
- What is Arnold s cat map, and why is it chaotic? provenance
- Which entry fits Arnold s cat map? action
Context Topology.
Context.
Covering spaces
Deck transformations.
Covering spaces
Covering spaces.
- What are deck transformations of covering spaces? provenance
- Which entry fits covering space? action
Symmetry
Symmetry concept.
Symmetry
Symmetry.
- How do automorphisms formalise symmetry? provenance
- Is the information accurate? boundary
What the second pass must settle
- Should graph automorphism be a separate entry?
- Should hyperbolic motion be moved to hyperbolic geometry entries?
- How should algebra references be linked?