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| 1 | +/- |
| 2 | +Copyright (c) 2021 Johan Commelin. All rights reserved. |
| 3 | +Released under Apache 2.0 license as described in the file LICENSE. |
| 4 | +Authors: Johan Commelin |
| 5 | +
|
| 6 | +! This file was ported from Lean 3 source module group_theory.subgroup.saturated |
| 7 | +! leanprover-community/mathlib commit f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c |
| 8 | +! Please do not edit these lines, except to modify the commit id |
| 9 | +! if you have ported upstream changes. |
| 10 | +-/ |
| 11 | +import Mathlib.GroupTheory.Subgroup.Basic |
| 12 | + |
| 13 | +/-! |
| 14 | +# Saturated subgroups |
| 15 | +
|
| 16 | +## Tags |
| 17 | +subgroup, subgroups |
| 18 | +
|
| 19 | +-/ |
| 20 | + |
| 21 | + |
| 22 | +namespace Subgroup |
| 23 | + |
| 24 | +variable {G : Type _} [Group G] |
| 25 | + |
| 26 | +/-- A subgroup `H` of `G` is *saturated* if for all `n : ℕ` and `g : G` with `g^n ∈ H` |
| 27 | +we have `n = 0` or `g ∈ H`. -/ |
| 28 | +@[to_additive |
| 29 | + "An additive subgroup `H` of `G` is *saturated* if for all `n : ℕ` and `g : G` with `n•g ∈ H` |
| 30 | + we have `n = 0` or `g ∈ H`."] |
| 31 | +def Saturated (H : Subgroup G) : Prop := |
| 32 | + ∀ ⦃n g⦄, g ^ n ∈ H → n = 0 ∨ g ∈ H |
| 33 | +#align subgroup.saturated Subgroup.Saturated |
| 34 | +#align add_subgroup.saturated AddSubgroup.Saturated |
| 35 | + |
| 36 | +@[to_additive] |
| 37 | +theorem saturated_iff_npow {H : Subgroup G} : |
| 38 | + Saturated H ↔ ∀ (n : ℕ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := |
| 39 | + Iff.rfl |
| 40 | +#align subgroup.saturated_iff_npow Subgroup.saturated_iff_npow |
| 41 | +#align add_subgroup.saturated_iff_nsmul AddSubgroup.saturated_iff_nsmul |
| 42 | + |
| 43 | +@[to_additive] |
| 44 | +theorem saturated_iff_zpow {H : Subgroup G} : |
| 45 | + Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by |
| 46 | + constructor |
| 47 | + · intros hH n g hgn |
| 48 | + induction' n with n n |
| 49 | + · simp only [Int.coe_nat_eq_zero, Int.ofNat_eq_coe, zpow_ofNat] at hgn⊢ |
| 50 | + exact hH hgn |
| 51 | + · suffices g ^ (n + 1) ∈ H by |
| 52 | + refine' (hH this).imp _ id |
| 53 | + simp only [IsEmpty.forall_iff, Nat.succ_ne_zero] |
| 54 | + simpa only [inv_mem_iff, zpow_negSucc] using hgn |
| 55 | + · intro h n g hgn |
| 56 | + specialize h n g |
| 57 | + simp only [Int.coe_nat_eq_zero, zpow_ofNat] at h |
| 58 | + apply h hgn |
| 59 | +#align subgroup.saturated_iff_zpow Subgroup.saturated_iff_zpow |
| 60 | +#align add_subgroup.saturated_iff_zsmul AddSubgroup.saturated_iff_zsmul |
| 61 | + |
| 62 | +end Subgroup |
| 63 | + |
| 64 | +namespace AddSubgroup |
| 65 | + |
| 66 | +theorem ker_saturated {A₁ A₂ : Type _} [AddCommGroup A₁] [AddCommGroup A₂] [NoZeroSMulDivisors ℕ A₂] |
| 67 | + (f : A₁ →+ A₂) : f.ker.Saturated := by |
| 68 | + intro n g hg |
| 69 | + simpa only [f.mem_ker, nsmul_eq_smul, f.map_nsmul, smul_eq_zero] using hg |
| 70 | +#align add_subgroup.ker_saturated AddSubgroup.ker_saturated |
| 71 | + |
| 72 | +end AddSubgroup |
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