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| | 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. |
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| | ==Definition== |
| | 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". |
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| | ==References== |
| | * {{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}} |
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| | [[Category:Subgroup properties]] |
| | [[Category:Abelian group theory]] |
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| | {{Abstract-algebra-stub}} |
Revision as of 13:04, 13 December 2013
In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept was generalized to essential submodules.
Definition
A subgroup of a (typically abelian) group 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".
References
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