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Theorems · Theorem · group theory

isSimpleGroup_iff

∀ (G : Type u_1) [inst : Group G], IsSimpleGroup G ↔ Nontrivial G ∧ ∀ (H : Subgroup G), H.Normal → H = ⊥ ∨ H = ⊤
Defined in
Mathlib.GroupTheory.Subgroup.Simple
Cited by
1 results in Mathlib
Foundations
Depth 18 from the axioms · uses propext, Quot.sound
Assumes
Group
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Around this declaration

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

Subgroup.isSimpleGroup_iff · cited by 0Subgroup.isSimpleGroup_iffTop.top · cited by 9680Top.topGroup · cited by 6238GroupBot.bot · cited by 4720Bot.botSubgroup · cited by 3593SubgroupNontrivial · cited by 2416NontrivialSubgroup.Normal · cited by 334Subgroup.NormalIsSimpleGroup · cited by 26IsSimpleGroupIsSimpleGroup.casesOn · cited by 1IsSimpleGroup.casesOnisSimpleGroup_iffCITED BYCITES

Cites8

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Cited by1

Results whose statement or proof uses this declaration.