Extensions of locally compact abelian, torsion-free groups by compact torsion abelian groups

Let $X$ be a compact torsion abelian group. In this paper, we show that an extension of $F_{p}$ by $X$ splits where $F_{p}$ is the p-adic number group and $p$ a prime number. Also, we show that an extension of a torsion-free, non-divisible LCA group by $X$ is not split.


Introduction
Throughout, all groups are Hausdorff abelian topological groups and will be written additively. Let £ denote the category of locally compact abelian (LCA) groups with continuous homomorphisms as morphisms. The Pontrjagin dual of a group G is denoted byĜ. A morphism is called proper if it is open onto its image and a short exact sequence 0 → A φ → B ψ → C → 0 in £ is said to be proper exact if φ and ψ are proper morphisms. In this case the sequence is called an extension of A by C ( in £). Following [5], we let Ext(C, A) denote the (discrete) group of extensions of A by C. In [4,Theorem 1], it is proved that if C is a compact torsion group and G a divisible LCA group, then Ext(C, G) = 0. However, the suggested proof in [4] appears to be incomplete as it uses the incorrect Proposition 8 of [2]. In [7], we proved that if G is σ−compact, then Ext(C, G) = 0. In this paper, we show that if G is a divisible, torsion-free LCA group, then Ext(C, G) = 0.
The additive topological group of real numbers is denoted by R, Q is the group of rationales with discrete topology and Z is the group of integers. For a prime number p, F p is the p-adic number groups which is the minimal divisible extension of J p . The topological isomorphism will be denote by " ∼ = ". For more on locally compact abelian groups see [6].

Main Results
Lemma 2.1. Let X ∈ £. Then nExt(X, F p ) = Ext(X, F p ) for every positive integer n.
Proof. It is sufficient to consider that by [ Proof. Since F p is totally disconnected, so it contains a compact open subgroup K. Consider the exact sequence 0 → K → F p → F p /K → 0. By [5, Corollary 2.10], there exists a short exact sequence Since F p is divisible, so by [5,Theorem 3.4] Ext(X, F p /K) = 0. On the other hand, X is torsion and F p torsion-free. Hence, Hom(X, F p ) = 0. So, we have the following exact sequence Since X is compact torsion, so nX = 0 for some n. Hence, nExt(X, K) = 0. Since (*) is exact, so nExt(X, F p ) = 0. Hence by Lemma1, Ext(X, F p ) = 0. Remark 2.3. Let X be a group. If f : X → X,f (x) = nx is topological isomorphism for each positive integer n, then X is a divisible, torsion-free group.
Lemma 2.4. Let X be a compact group. Then Ext(X, F p ) is a divisible, torsion-free group.
Proof. Let n be an arbitrary positive integer. Then the exact sequence 0 → X ×n → X → X/nX → 0 induces the following exact sequence Since H/K is a discrete group and F p a divisible group, so Ext(H/K, F p ) = 0. Hence Ext(H, F p ) ∼ = Ext(K, F p ). By Lemma 2.4, Ext(K, F p ) is a divisible, torsion-free group. So Ext(X, F p ) is a divisible, torsion-free group.
Theorem 2.6 Let X be a compact torsion group and G a divisible, torsion-free group. Then Ext(X, G) = 0.
Theorem 2.8. Let X be a compact torsion group and G a torsion-free group. Then Ext(X, G) = 0.