Math 2.0

example

Action by left multiplication

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.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

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Sources

  1. Web https://en.wikipedia.org/wiki/Group_action, Examples

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