feat: port GroupTheory.Subgroup.Saturated (#1884)

Co-authored-by: Moritz Firsching <firsching@google.com>

Mathlib.lean

@@ -579,6 +579,7 @@ import Mathlib.GroupTheory.Perm.ViaEmbedding
 import Mathlib.GroupTheory.Subgroup.Actions
 import Mathlib.GroupTheory.Subgroup.Basic
 import Mathlib.GroupTheory.Subgroup.MulOpposite
+import Mathlib.GroupTheory.Subgroup.Saturated
 import Mathlib.GroupTheory.Subgroup.Simple
 import Mathlib.GroupTheory.Subgroup.Zpowers
 import Mathlib.GroupTheory.Submonoid.Basic

Mathlib/GroupTheory/Subgroup/Saturated.lean

@@ -0,0 +1,72 @@
+/-
+Copyright (c) 2021 Johan Commelin. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johan Commelin
+
+! This file was ported from Lean 3 source module group_theory.subgroup.saturated
+! leanprover-community/mathlib commit f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.GroupTheory.Subgroup.Basic
+
+/-!
+# Saturated subgroups
+
+## Tags
+subgroup, subgroups
+
+-/
+
+
+namespace Subgroup
+
+variable {G : Type _} [Group G]
+
+/-- A subgroup `H` of `G` is *saturated* if for all `n : ℕ` and `g : G` with `g^n ∈ H`
+we have `n = 0` or `g ∈ H`. -/
+@[to_additive
+      "An additive subgroup `H` of `G` is *saturated* if for all `n : ℕ` and `g : G` with `n•g ∈ H`
+      we have `n = 0` or `g ∈ H`."]
+def Saturated (H : Subgroup G) : Prop :=
+  ∀ ⦃n g⦄, g ^ n ∈ H → n = 0 ∨ g ∈ H
+#align subgroup.saturated Subgroup.Saturated
+#align add_subgroup.saturated AddSubgroup.Saturated
+
+@[to_additive]
+theorem saturated_iff_npow {H : Subgroup G} :
+    Saturated H ↔ ∀ (n : ℕ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H :=
+  Iff.rfl
+#align subgroup.saturated_iff_npow Subgroup.saturated_iff_npow
+#align add_subgroup.saturated_iff_nsmul AddSubgroup.saturated_iff_nsmul
+
+@[to_additive]
+theorem saturated_iff_zpow {H : Subgroup G} :
+    Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by
+  constructor
+  · intros hH n g hgn
+    induction' n with n n
+    · simp only [Int.coe_nat_eq_zero, Int.ofNat_eq_coe, zpow_ofNat] at hgn⊢
+      exact hH hgn
+    · suffices g ^ (n + 1) ∈ H by
+        refine' (hH this).imp _ id
+        simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]
+      simpa only [inv_mem_iff, zpow_negSucc] using hgn
+  · intro h n g hgn
+    specialize h n g
+    simp only [Int.coe_nat_eq_zero, zpow_ofNat] at h
+    apply h hgn
+#align subgroup.saturated_iff_zpow Subgroup.saturated_iff_zpow
+#align add_subgroup.saturated_iff_zsmul AddSubgroup.saturated_iff_zsmul
+
+end Subgroup
+
+namespace AddSubgroup
+
+theorem ker_saturated {A₁ A₂ : Type _} [AddCommGroup A₁] [AddCommGroup A₂] [NoZeroSMulDivisors ℕ A₂]
+    (f : A₁ →+ A₂) : f.ker.Saturated := by
+  intro n g hg
+  simpa only [f.mem_ker, nsmul_eq_smul, f.map_nsmul, smul_eq_zero] using hg
+#align add_subgroup.ker_saturated AddSubgroup.ker_saturated
+
+end AddSubgroup