Theorems · Theorem · group theory
Equiv.Perm.IsThreeCycle.eq_swap_mul_swap_iff_mem_support
∀ {α : Type u_1} [inst : Fintype α] [inst_1 : DecidableEq α] {g : Equiv.Perm α} {a : α},
g.IsThreeCycle → (g = Equiv.swap a (g a) * Equiv.swap (g a) (g (g a)) ↔ a ∈ g.support)- Defined in
- Mathlib.GroupTheory.Perm.Cycle.Type
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Finsetstatement · cited by 13,712
- Fintypestatement and proof · cited by 7,736
- Equiv.Permstatement and proof · cited by 1,375
- Equiv.Perm.supportstatement and proof · cited by 230
- Equiv.swapstatement and proof · cited by 197
- Equiv.Perm.extproof · cited by 75
- Equiv.Perm.mem_supportproof · cited by 51
- Equiv.swap_apply_leftproof · cited by 44
- Equiv.swap_apply_of_ne_of_neproof · cited by 39
- Equiv.Perm.IsThreeCyclestatement and proof · cited by 34
- Equiv.swap_apply_rightproof · cited by 31
Cited by1
Results whose statement or proof uses this declaration.
- Equiv.Perm.closure_cycleType_eq_two_two_eq_alternatingGroupproof · cited by 2