Theorems · Theorem · group theory
closure_of_isSwap_of_isPretransitive
∀ {α : Type u_2} [inst : DecidableEq α] [Finite α] {S : Set (Equiv.Perm α)},
(∀ σ ∈ S, σ.IsSwap) → ∀ [MulAction.IsPretransitive (↥(Subgroup.closure S)) α], Subgroup.closure S = ⊤A transitive permutation group generated by transpositions must be the whole symmetric group
- Defined in
- Mathlib.GroupTheory.Perm.ClosureSwap
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Top.topstatement · cited by 9,680
- Subgroupstatement · cited by 3,593
- Finitestatement and proof · cited by 3,029
- Compl.complproof · cited by 2,925
- Equiv.Permstatement and proof · cited by 1,375
- Subgroup.closurestatement and proof · cited by 196
- MulAction.IsPretransitivestatement and proof · cited by 94
- MulAction.fixedByproof · cited by 36
- Equiv.Perm.IsSwapstatement and proof · cited by 35
- mem_closure_isSwapproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- surjective_of_isSwap_of_isPretransitive'proof · cited by 2