On T-Extensions of Abelian Groups

Document Type : Original Article

Authors

Aliakbar Alijani ¹

Hossein Sahleh ²

¹ Mollasadra Technical and Vocational College, Technical and Vocational University, Ramsar, Iran.

² Department of Mathematics, University of Guilan, P. O. Box 1914, Rasht, Iran.

10.22034/kjm.2018.74220

Abstract

Let $\Re$ be the category of all discrete abelian groups, and let $\cal{L}$ be the category of all locally compact abelian (LCA) groups. For a group $G\in \cal{L}$, the maximal torsion subgroup of $G$ is denoted by $tG$. A short exact sequence $0\to A\stackrel{\phi}{\to} B\stackrel{\psi}{\to}C\to 0$ in $\Re$ is said to be a t-extension if $0\to tA\stackrel{\phi}{\to} tB\stackrel{\psi}{\to}tC\to 0$ is a short exact sequence. We show that the set of all t-extensions of $A$ by $C$ is a subgroup of $Ext(C,A)$, which contains $Pext(C,A)$ for discrete abelian groups $A$ and $C$. We establish conditions under which the t-extensions split and determine those groups in $\Re$ which are t-injective or t-projective in $\Re$. Finally we determine the compact groups $G$ in $\cal{L}$ such that every pure extension of $G$ by a compact connected group $C\in \cal{L}$ splits.

Keywords

T-extensions

extensions

pure extensions

locally compact abelian groups

Subjects

• 20 Group theory and generalizations
• 22 Topological groups, Lie groups

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