orbit-stabilizer theorem
E620661
The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
result in group theory
ⓘ
theorem ⓘ |
| appliesTo |
finite groups
ⓘ
group actions on sets ⓘ permutation representations ⓘ |
| area | abstract algebra ⓘ |
| assumes | group action of G on X ⓘ |
| classification | fundamental theorem of group actions ⓘ |
| defines |
orbit of x as {g·x : g in G}
ⓘ
stabilizer of x as {g in G : g·x = x} ⓘ |
| field | group theory ⓘ |
| formalStatement | For a group G acting on a set X and x in X, |G| = |Orb(x)| · |Stab(x)| when all sets are finite ⓘ |
| generalizes | Lagrange's theorem for transitive actions ⓘ |
| holdsFor |
actions by conjugation
ⓘ
actions by permutations ⓘ left group actions ⓘ right group actions ⓘ |
| implies |
index of stabilizer equals size of orbit
ⓘ
|G : Stab(x)| = |Orb(x)| for finite groups ⓘ |
| importance |
basic result taught in undergraduate algebra courses
ⓘ
central tool in finite group theory ⓘ |
| involves |
cardinality of sets
ⓘ
coset decomposition ⓘ group action ⓘ orbit ⓘ stabilizer subgroup ⓘ subgroup ⓘ |
| relatedTo |
Burnside's lemma
NERFINISHED
ⓘ
Cauchy's theorem NERFINISHED ⓘ Sylow theorems NERFINISHED ⓘ class equation NERFINISHED ⓘ coset decomposition theorem ⓘ |
| relates |
size of a group
ⓘ
size of a stabilizer subgroup ⓘ size of an orbit ⓘ |
| usedFor |
Burnside's lemma proofs
ⓘ
analyzing group actions on geometric objects ⓘ class equation derivation ⓘ computing orbit sizes ⓘ computing stabilizer sizes ⓘ counting arguments in combinatorics ⓘ proving Lagrange's theorem ⓘ studying symmetry groups ⓘ |
| usedIn |
Galois theory
NERFINISHED
ⓘ
algebraic combinatorics ⓘ algebraic geometry group actions ⓘ representation theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.