Math 2.0

Group theory and generalizations Definition

Faithful action

Builds on Group actionKernel of a homomorphism

Definition

Let a group GG act on a set XX. The action is faithful when, for each g∈Gg \in G, g⋅x=xg \cdot x = x for all x∈Xx \in X implies g=eg = e.

Remarks

The action is faithful if and only if the associated homomorphism G→Sym⁡(X)G \to \operatorname{Sym}(X) is injective. So a faithful action identifies GG with a subgroup of Sym⁡(X)\operatorname{Sym}(X). The action of GG on itself by conjugation is faithful if and only if the centre of GG is trivial.

Reading path

To understand this page from the ground up, read these 6 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 actiondefinition
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.
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  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

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_action, Notable properties of actions
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MSC20B05
DefinesFaithful action
UsesGroup action, Kernel of a homomorphism

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