Commutator Invariant Subgroups of Abelian Groups

Abstract

We describe the commutator invariant subgroups of a nonreduced abelian group. We find out when all commutator invariant subgroups of a separable group and an algebraically compact torsionfree group are fully invariant and describe the E-centers and E-commutants of these and some other groups.Let A be an abelian group. Denote the endomorphism ring of A by E(A), put A 1 = ∞ n=1 nA, and let r(A) be the rank of A. Unless otherwise specified, A p is the p-component, t(A) is the periodic part, andIf a is an element of order p k then by e(a) = k we denote the exponent of a. The expression H A means that H is a subgroup in A, and the expression H fiA, that H is a fully invariant subgroup in A, i.e., ϕH ⊆ H for every ϕ ∈ E(A). (B,G) fX be the subgroup in G generated by all homomorphic images of X. Denote by 1 A the identity endomorphism of a group A. If A is a homogeneous torsion-free group then t(A) is the type of A, N is the set of all natural numbers, Z is the ring or the group of integers, Q is the field of all rationals, Z p is the ring or the group of p-adic integers, and P is the set of all primes.AIf R is a ring then the operation ϕ • ψ = ϕψ − ψϕ (where ϕ, ψ ∈ R) is called commutation and the element [ϕ, ψ] = ϕψ − ψϕ is called the commutator of ϕ and ψ.The following are verified directly:(1) [α, β]γ = α[β, γ] + [αγ, β], [α, β]γ = [α, βγ] + β[γ, α]; (2) γ[α, β] = [γ, α]β + [α, γβ], γ[α, β] = [γα, β] + [β, γ]α; (3) in the ring R, the operation of commutation is associative if and only if every commutator of a ring R lies in the center of R. Indeed, [[a, b], c] = abc − bac − cab + cba, [a, [b, c]] = abc − acb − bca + cba. Equating the right-hand sides, we infer 0 = bac + cab − acb − bca = [[c, a], b], whence [c, a] ∈ Z(R). Call a subgroup H A commutator invariant (briefly, a ci-subgroup) if [ϕ, ψ]h ∈ H for all h ∈ H and ϕ, ψ ∈ E(A). Each subgroup in a group with commutative endomorphism ring is a ci-subgroup. Lemma 1. Let A = i∈I A i and let π i : A → A i be the corresponding projections and H A. Then (1) H ciA if and only if Hom(and only if Hom(A i , A j )B i ⊆ B j for all i, j ∈ I, where j = i; (3) if A i fiA and B i A i then B = i∈I B i ciA if and only if B i ciA i for all i ∈ I; (4) a ci-subgroup H of A is its fi-subgroup if and only if π i H = H ∩ A i and H ∩ A i fiA i for every i ∈ I.