Math 2.0

Group theory and generalizations Definition

Automorphism group

Builds on Group isomorphism

Definition

Let GG be a group. An automorphism of GG is a group isomorphism from GG to GG. The automorphism group Aut⁡(G)\operatorname{Aut}(G) is the set of all automorphisms of GG, with composition of maps as the operation.

Remarks

The identity map is the identity element of Aut⁡(G)\operatorname{Aut}(G), and the inverse of an automorphism is its inverse map. The automorphism group of (ℤ,+)(\mathbb{Z}, +) has two elements, the identity and n↦−nn \mapsto -n. For an abelian group GG, the map g↦g−1g \mapsto g^{-1} is an automorphism.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionGroup isomorphism, definition, no evidence levelGroup isomorphismdefinitionAutomorphism group, definition, no evidence levelAutomorphism groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Group homomorphism definition
  3. Group isomorphism definition
  4. Automorphism group definition

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Automorphism_group, lead section
About this page
Tag0CWXCWP0Y0
URIm2://0CWXCWP0Y0
Aliasesgroup-of-automorphisms
Kinddefinition
WritingW1 stated
MSC20F28
DefinesAutomorphism group
UsesGroup isomorphism

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by