Math 2.0

Group theory and generalizations Example

Action by left multiplication

Builds on Group actionTransitive actionFaithful action

Example

Let GG be a group. The map G×G→GG \times G \to G, (g,x)↦gx(g, x) \mapsto gx, is an action of GG on the set GG. The two axioms are ex=xex = x and g(hx)=(gh)xg(hx) = (gh)x, and they follow from the group axioms. This action is transitive, because (yx−1)x=y(yx^{-1})x = y for all x,y∈Gx, y \in G. The stabiliser of each point is {e}\{e\}, because gx=xgx = x implies g=eg = e. In particular, the action is faithful, and this gives Cayley's theorem.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionGroup action, definition, no evidence levelGroup actiondefinitionFaithful action, definition, no evidence levelFaithful actiondefinitionOrbit, definition, no evidence levelOrbitdefinitionTransitive action, definition, no evidence levelTransitive actiondefinitionAction by left multiplication, example, no evidence levelAction by left multip…example
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. Symmetric group definition
  4. Kernel of a homomorphism definition
  5. Group action definition
  6. Faithful action definition
  7. Orbit definition
  8. Transitive action definition
  9. Action by left multiplication example

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_action, Examples
About this page
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Instance ofGroup action
UsesGroup action, Transitive action, Faithful action

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