https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=161.51.11.2formulasearchengine - User contributions [en]2026-07-28T12:45:01ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Leak_detection&diff=22812Leak detection2013-12-13T12:04:22Z<p>161.51.11.2: </p>
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<div>In [[mathematics]], especially in the area of [[abstract algebra|algebra]] studying the theory of [[abelian group]]s, an '''essential subgroup''' is a subgroup that determines much of the structure of its containing group. The concept was generalized to [[essential submodule]]s.<br />
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==Definition==<br />
A [[subgroup]] <math>S</math> of a (typically [[abelian group|abelian]]) [[Group (mathematics)|group]] <math>G</math> is said to be '''essential''' if whenever ''H'' is a non-trivial subgroup of ''G'', the intersection of ''S'' and ''H'' is non-trivial: here "non-trivial" means "containing an element other than the identity".<br />
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==References==<br />
* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}}<br />
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[[Category:Subgroup properties]]<br />
[[Category:Abelian group theory]]<br />
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{{Abstract-algebra-stub}}</div>161.51.11.2