Theorems · Theorem · group theory

alternatingGroup.stabilizer_isPreprimitive

∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] {s : Set α},
  sᶜ.Nontrivial → MulAction.IsPreprimitive ↥(MulAction.stabilizer (↥(alternatingGroup α)) s) ↑s

In the alternating group, the stabilizer of a set acts primitively on that set if the complement is nontrivial.

Defined in: Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups
Cited by: 1 results in Mathlib
Foundations: Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Fintype DecidableEq

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites13

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
Fintypestatement and proof · cited by 7,736
Set.Elemstatement · cited by 7,166
Subgroupstatement · cited by 3,593
Compl.complstatement and proof · cited by 2,925
Equiv.Permstatement · cited by 1,375
MulAction.stabilizerstatement · cited by 254
Set.Nontrivialstatement and proof · cited by 145
alternatingGroupstatement · cited by 96
Set.mulActionSetstatement · cited by 95
MulAction.IsPreprimitivestatement · cited by 50
alternatingGroup.stabilizer.surjective_toPermproof · cited by 2

Cited by1

Results whose statement or proof uses this declaration.

alternatingGroup.stabilizer_subgroup_isPreprimitiveproof · cited by 1