Archimedean groups #

This file proves a few facts about ordered groups which satisfy the archimedean property, that is: class archimedean (α) [ordered_add_comm_monoid α] : Prop := (arch : ∀ (x : α) {y}, 0 < y → ∃ n : ℕ, x ≤ n • y)

They are placed here in a separate file (rather than incorporated as a continuation of algebra.order.archimedean) because they rely on some imports from group_theory -- bundled subgroups in particular.

The main result is add_subgroup.cyclic_of_min: a subgroup of a decidable archimedean abelian group is cyclic, if its set of positive elements has a minimal element.

This result is used in this file to deduce int.subgroup_cyclic, proving that every subgroup of ℤ is cyclic. (There are several other methods one could use to prove this fact, including more purely algebraic methods, but none seem to exist in mathlib as of writing. The closest is subgroup.is_cyclic, but that has not been transferred to add_subgroup.)

The result is also used in topology.instances.real as an ingredient in the classification of subgroups of ℝ.

source

theorem add_subgroup.cyclic_of_min {G : Type u_1} [linear_ordered_add_comm_group G] [archimedean G] {H : add_subgroup G} {a : G} (ha : is_least {g : G | g ∈ H ∧ 0 < g} a) :

H = add_subgroup.closure {a}

Given a subgroup H of a decidable linearly ordered archimedean abelian group G, if there exists a minimal element a of H ∩ G_{>0} then H is generated by a.

source

theorem int.subgroup_cyclic (H : add_subgroup ℤ) :

∃ (a : ℤ), H = add_subgroup.closure {a}

Every subgroup of ℤ is cyclic.