Group theory and generalizations Definition
Transitive action
Builds on Group actionOrbit
Definition
Let a group act on a nonempty set . The action is transitive when for all there is a with .
Remarks
The action is transitive if and only if it has exactly one orbit. The action of on is transitive. The action of on the left cosets of a subgroup is transitive. Each transitive action of is isomorphic to such an action.
Reading path
To understand this page from the ground up, read these 6 pages in order. The foundations come first.
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- no evidence level
- Group concept
- Group homomorphism definition
- Symmetric group definition
- Group action definition
- Orbit definition
- Transitive action definition
Where this leads
28 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Cayley's theorem
- Finite p-groups are nilpotent
- Class equation
- Fixed points of a p-group action
- Second Sylow theorem
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- Action on left cosets
- Orbit-stabiliser theorem
- Abelian simple groups have prime order
- Cauchy's theorem
- Action by left multiplication
- The number of Sylow subgroups divides the index
- Third Sylow theorem
- Frattini argument
- A nontrivial finite p-group has nontrivial centre
- Size of an orbit times the size of the stabiliser
- First Sylow theorem
- Size of a conjugacy class is the index of the centraliser
- Groups of order pq
- Burnside's lemma
- Galois group of a prime degree polynomial with two nonreal roots
- Hall's theorem on solvable groups
- Permutation group
- Hall subgroup
- Sylow theorems
- Frattini subgroup
Further reading
- Web https://en.wikipedia.org/wiki/Group_action, Transitivity properties
About this page
| Tag | 0QP2YYR79N |
|---|---|
| URI | m2://0QP2YYR79N |
| Aliases | transitive-group-action |
| Kind | definition |
| Writing | W0 stub |
| MSC | 20B05 |
| Defines | Transitive action |
| Uses | Group action, Orbit |
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