en>Funandtrvl: nvbx

2013-06-28T23:24:27Z

nvbx

← Older revision		Revision as of 01:24, 29 June 2013
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	~~Friends contact her Claude Gulledge. Interviewing is what she does~~ in ~~her day occupation but soon her husband and her will start their own company. Delaware is~~ the ~~only place I~~'~~ve been residing~~ in. ~~What she loves performing~~ is to ~~perform croquet but she hasn~~'~~t made a dime with~~ it.<br><br>~~Look into my site ::~~ [http://~~Ghaziabadmart~~.~~com~~/~~oxwall~~/~~blogs~~/~~post~~/~~16946~~ http://~~Ghaziabadmart~~.~~com~~/~~oxwall~~/~~blogs~~/~~post~~/~~16946~~]		In [[mathematics]], especially in the area of [[abstract algebra\|algebra]] studying the theory of [[abelian group]]s, a '''pure subgroup''' is a generalization of [[direct summand]]. It has found many uses in abelian group theory and related areas.

			==Definition==
			A [[subgroup]] <math>S</math> of a (typically [[abelian group\|abelian]]) [[Group (mathematics)\|group]] <math>G</math> is said to be '''pure''' if whenever an element of <math>S</math> has an <math>n^{th}</math> root in <math>G</math>, it necessarily has an <math>n^{th}</math> root in <math>S</math>.

			==Origins==
			Pure subgroups are also called '''isolated subgroups''' or '''serving subgroups''' and were first investigated in [[Heinz Prüfer\|Prüfer]]'s 1923 paper<ref>{{cite journal \| title = Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen \| first = H. \| last = Prüfer \| journal = Math. Zeit. \| volume = 17 \| issue = 1 \| pages=35–61 \| year = 1923 \| doi = 10.1007/BF01504333 \| url = http://www.zentralblatt-math.org/zmath/en/search/?q=an:49.0084.03&format=complete \| format = }} {{dead link\|date=March 2010}}</ref> which described conditions for the decomposition of primary [[abelian group]]s as [[direct sum of abelian groups\|direct sum]]s of [[cyclic group]]s using pure subgroups. The work of Prüfer was complemented by Kulikoff<ref>{{cite journal \| title = Zur Theorie der Abelschen Gruppen von beliebiger Mächtigkeit. \| first = L. \| last = Kulikoff \| journal = Rec. Math. Moscou, n. Ser. \| volume = 9 \| pages = 165–181 \| year = 1941 \| url = http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0025.29901&format=complete \| format = }} {{dead link\|date=March 2010}}</ref> where many results were proved again using pure subgroups systematically. In particular, a proof was given that pure subgroups of finite exponent are direct summands. A more complete discussion of pure subgroups, their relation to [[Abelian group#Infinite abelian groups\|infinite abelian group]] theory, and a survey of their literature is given in [[Irving Kaplansky]]'s little red book.<ref>{{cite book \| title = Infinite Abelian Groups \| first = Irving \| last = Kaplansky \| publisher = University of Michigan \| year = 1954 \| authorlink1 = Irving Kaplansky \| isbn = 0-472-08500-X }}</ref>

			==Examples==
			* Every direct summand of a group is a pure subgroup
			* Every pure subgroup of a pure subgroup is pure.
			* A [[divisible group\|divisible]] subgroup of an Abelian group is pure.
			* If the quotient group is torsion-free, the subgroup is pure.
			* The torsion subgroup of an Abelian group is pure.
			* The union of pure subgroups is a pure subgroup.

			Since in a finitely generated Abelian group the torsion subgroup is a direct summand, one might ask if the torsion subgroup is always a direct summand of an Abelian group. It turns out that it is not always a summand, but it ''is'' a pure subgroup. Under certain mild conditions, pure subgroups are direct summands. So, one can still recover the desired result under those conditions, as in Kulikoff's paper. Pure subgroups can be used as an intermediate property between a result on direct summands with finiteness conditions and a full result on direct summands with less restrictive finiteness conditions. Another example of this use is Prüfer's paper, where the fact that "finite torsion Abelian groups are direct sums of cyclic groups" is extended to the result that "all torsion Abelian groups of finite [[exponent (group theory)\|exponent]] are direct sums of cyclic groups" via an intermediate consideration of pure subgroups.

			==Generalizations==

			Pure subgroups were generalized in several ways in the theory of abelian groups and modules. [[Pure submodule]]s were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots. [[Pure injective module\|Pure injective]] and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is a direct summand of a direct sum of finitely presented modules. In the case of the integers and Abelian groups a pure projective module amounts to a direct sum of cyclic groups.

			==References==
			<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 \| pages=9–16}} Chapter III.

			[[Category:Subgroup properties]]
			[[Category:Abelian group theory]]

en>Jason Davies: Fix grammar

2010-07-24T21:36:19Z

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Friends contact her Claude Gulledge. Interviewing is what she does in her day occupation but soon her husband and her will start their own company. Delaware is the only place I've been residing in. What she loves performing is to perform croquet but she hasn't made a dime with it.<br><br>Look into my site :: [http://Ghaziabadmart.com/oxwall/blogs/post/16946 http://Ghaziabadmart.com/oxwall/blogs/post/16946]

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