Left group actions, transitive actions, and faithful actions

Depends on

• Group and abelian group

Used by

• A transitive action is faithful exactly when a point stabiliser is core-free Corollary
• G-sets are strictly monadic over sets Corollary
• The action of ℤ/2 on two disjoint two-point orbits is free but not transitive Counterexample
• The natural action of S₃ on three points is faithful and transitive but not free Counterexample
• A free group action has no nonidentity element fixing a point Definition
• A group acting on a ring by automorphisms and its invariant subring Definition
• An action of a group H on a group N by automorphisms Definition
• An R-linear action of G on a left R-module, and a G-module over R Definition
• Blocks and block systems for a group action Definition
• Colourings, weight functions, and the pattern inventory Definition
• Covering-space actions by disjoint translates of neighbourhoods Definition
• Crossed homomorphism for a G-group Definition
• Deck transformations and the deck-transformation group of a covering Definition
• Equivariant maps and isomorphisms of group actions Definition
• Graph automorphisms and group actions on a simplicial graph Definition
• Isometric, proper, and cobounded actions on metric spaces Definition
• k-transitive and k-homogeneous actions Definition
• Permutation groups, stabilizers, supports, and normal filters Definition
• Primitive and imprimitive transitive actions Definition
• Regular actions Definition
• The cycle construction CYC(A) Definition
• The cycle index of a finite permutation group Definition
• The fixed-point sets Xᵍ and X^G of a group action Definition
• The imprimitive wreath product of permutation groups Definition
• The monodromy right action on a covering fibre and its equivalent left-action convention Definition
• The orbit G· x and stabilizer Gₓ of a point in a group action Definition
• The trivial representation, the regular representation, and permutation representations from finite G-sets Definition
• An action groupoid has the acted-on set as objects, orbits as connected components, and stabilizers as automorphism groups Example
• For a group G the monad G×(-) on sets has the G-sets as its algebras Example
• Frobenius reciprocity for group representations without tensor products Example
• Left multiplication gives a free and transitive action of every group on itself Example
• The four rotations of a square act freely, transitively and faithfully on its vertices Example
• The orbit-set and fixed-point constructions as Kan extensions Example
• The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
• The trivial action of ℤ/2 on a singleton is transitive but not faithful Example
• There are six binary necklaces of length four up to rotation Example
• There are six two-colourings of the vertices of a square up to its eight symmetries Example
• When the group order is invertible the Reynolds operator retracts a ring onto its invariants Example
• Cyclic shifting is an action of ℤ/m on the words of length m over a set Lemma
• For a finite group of ring automorphisms the orbit polynomial is monic over the invariant subring, so the ring is integral over its invariants Lemma

…and 20 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Brosnan, Group actions (standard reference, not scraped)