feat(group_theory/subgroup/basic): If H is commutative, then H ≤ H.centralizer (#16718)

If `H` is commutative, then `H ≤ H.centralizer`.

Author: tb65536

src/group_theory/subgroup/basic.lean

@@ -1859,6 +1859,12 @@ end⟩⟩
   (H.subgroup_of K).is_commutative :=
 H.comap_injective_is_commutative subtype.coe_injective
 
+@[to_additive] lemma le_centralizer_iff_is_commutative : K ≤ K.centralizer ↔ K.is_commutative :=
+⟨λ h, ⟨⟨λ x y, subtype.ext (h y.2 x x.2)⟩⟩, λ h x hx y hy, congr_arg coe (h.1.1 ⟨y, hy⟩ ⟨x, hx⟩)⟩
+
+@[to_additive] lemma le_centralizer [h : H.is_commutative] : H ≤ H.centralizer :=
+le_centralizer_iff_is_commutative.mpr h
+
 end subgroup
 
 namespace group