English
In a 2-transitive action, the normal closure of a stabilizer equals the whole group.
Русский
В действии 2-пер(tm)иводном действии нормальное замыкание стабилизатора равно всей группе.
LaTeX
$$\operatorname{normalClosure}(\operatorname{stabilizer}_G(a)) = G$$
Lean4
/-- In a 2-transitive action, the normal closure of stabilizers is the full group. -/
theorem normalClosure_of_stabilizer_eq_top (hsn' : 2 < ENat.card α) (hG' : IsMultiplyPretransitive G α 2) {a : α} :
normalClosure ((stabilizer G a) : Set G) = ⊤ :=
by
have : IsPretransitive G α := by
rw [← is_one_pretransitive_iff]
exact isMultiplyPretransitive_of_le' (one_le_two) (le_of_lt hsn')
have : Nontrivial α := by
rw [← ENat.one_lt_card_iff_nontrivial]
exact lt_trans (by norm_num) hsn'
have hGa : IsCoatom (stabilizer G a) :=
by
rw [isCoatom_stabilizer_iff_preprimitive]
exact isPreprimitive_of_is_two_pretransitive hG'
apply hGa.right
constructor
· apply le_normalClosure
· intro hyp
have : Nontrivial (ofStabilizer G a) :=
by
rw [← ENat.one_lt_card_iff_nontrivial]
apply lt_of_add_lt_add_right
rwa [ENat_card_ofStabilizer_add_one_eq]
rw [nontrivial_iff] at this
obtain ⟨b, c, hbc⟩ := this
have : IsPretransitive (stabilizer G a) (ofStabilizer G a) :=
by
rw [← is_one_pretransitive_iff]
rwa [← ofStabilizer.isMultiplyPretransitive]
-- get g ∈ stabilizer G a, g • b = c,
obtain ⟨⟨g, hg⟩, hgbc⟩ := exists_smul_eq (stabilizer G a) b c
apply hbc
rw [← SetLike.coe_eq_coe] at hgbc ⊢
obtain ⟨h, hinvab⟩ := exists_smul_eq G (b : α) a
rw [eq_comm, ← inv_smul_eq_iff] at hinvab
rw [← hgbc, SetLike.val_smul, ← hinvab, inv_smul_eq_iff, eq_comm]
simp only [subgroup_smul_def, smul_smul, ← mul_assoc, ← mem_stabilizer_iff]
exact
hyp
(normalClosure_normal.conj_mem g (le_normalClosure hg) h)
-- Wielandt claims that this is proved by the same method as above.
