Theorems · Theorem · group theory

Equiv.Perm.alternatingGroup_le_of_normal

∀ {α : Type u_2} [inst : DecidableEq α] [inst_1 : Fintype α],
  5 ≤ Nat.card α → ∀ {N : Subgroup (Equiv.Perm α)} [N.Normal], Nontrivial ↥N → alternatingGroup α ≤ N

If α has at least 5 elements, then any nontrivial normal subgroup of Equiv.Perm α contains alternatingGroup α.

Defined in: Mathlib.GroupTheory.SpecificGroups.Alternating.Simple
Cited by: 0 results in Mathlib
Foundations: Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: DecidableEq Fintype Subgroup.Normal

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites15

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

Fintypestatement and proof · cited by 7,736
Set.Elemproof · cited by 7,166
Subgroupstatement and proof · cited by 3,593
Nontrivialstatement and proof · cited by 2,416
Equiv.Permstatement and proof · cited by 1,375
Nat.cardstatement and proof · cited by 844
Subgroup.Normalstatement and proof · cited by 334
Set.powersetCardproof · cited by 100
alternatingGroupstatement · cited by 96
MulAction.IsPreprimitiveproof · cited by 50
MulAction.IwasawaStructure.commutator_leproof · cited by 4
Set.powersetCard.fixedPoints_ne_univ_of_faithfulSMulproof · cited by 3

Cited by0

Results whose statement or proof uses this declaration.

Nothing cites this yet.