http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Algebraically_compact_group 2026-05-10T19:21:21Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Algebraically_compact_group&diff=3059768&oldid=prev 2023-08-12T23:27:44Z

<p><span class="autocomment">External links</span></p> <p><b>New page</b></p><div>In [[mathematics]], in the realm of [[Abelian group|abelian]] [[group theory]], a [[Group (mathematics)|group]] is said to be &#039;&#039;&#039;algebraically compact&#039;&#039;&#039; if it is a [[direct summand]] of every abelian group containing it as a [[pure subgroup]].<br /> <br /> Equivalent characterizations of algebraic compactness:<br /> * The reduced part of the group is Hausdorff and complete in the &lt;math&gt;\mathbb{Z}&lt;/math&gt; adic topology.<br /> * The group is &#039;&#039;pure injective&#039;&#039;, that is, injective with respect to exact sequences where the embedding is as a pure subgroup.<br /> <br /> Relations with other properties:<br /> * A [[torsion-free group]] is [[cotorsion group|cotorsion]] if and only if it is algebraically compact.<br /> * Every [[injective group]] is algebraically compact.<br /> * [[Ulm factor]]s of cotorsion groups are algebraically compact.<br /> ==External links==<br /> * [https://doi.org/10.1023%2FA%3A1020582507609 On endomorphism rings of Abelian groups]<br /> <br /> [[Category:Abelian group theory]]<br /> [[Category:Properties of groups]]<br /> <br /> <br /> {{group-theory-stub}}</div>

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