en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q780099

2013-03-16T16:46:10Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q780099

← Older revision		Revision as of 18:46, 16 March 2013
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	~~The author~~ is ~~known as Araceli Gulledge~~. I'~~ve always cherished residing~~ in ~~Idaho~~. ~~The thing she adores most is to perform handball but she can~~'~~t make it her occupation~~. ~~Bookkeeping has been his day job for a while~~.<br><br>~~My blog;~~ [~~http~~://~~Www~~.~~Creativecats.com~~/~~UserProfile~~/~~tabid~~/~~331~~/~~userId~~/~~19522~~/~~Default~~.~~aspx Www.Creativecats.com~~]		In [[algebra]], '''Hall's universal group''' is
			a countable [[locally finite group]], say ''U'', which is uniquely
			characterized by the following properties.

			* Every finite group ''G'' admits a [[monomorphism]] to ''U''.

			* All such monomorphisms are conjugate by [[inner automorphism]]s of ''U''.

			It was defined by [[Philip Hall]] in 1959.<ref>Hall, P.
			''Some constructions for locally finite groups.''
			J. London Math. Soc. '''34''' (1959) 305--319. {{MathSciNet \| id = 162845}}</ref>

			== Construction ==

			Take any group <math> \Gamma_0 </math> of order <math> \geq 3 </math>.
			Denote by <math> \Gamma_1 </math> the group <math> S_{\Gamma_0}</math>
			of [[permutation]]s of elements of <math> \Gamma_0 </math>, by
			<math>\Gamma_2 </math> the group

			:<math> S_{\Gamma_1}= S_{S_{\Gamma_0}} \, </math>

			and so on. Since a group acts faithfully on itself by permutations

			:<math> x\mapsto gx \, </math>

			according to [[Cayley's theorem]], this gives a chain of monomorphisms

			:<math>\Gamma_0 \hookrightarrow \Gamma_1 \hookrightarrow \Gamma_2 \hookrightarrow \cdots . \, </math>

			A [[direct limit]] (that is, a union) of all <math> \Gamma_i</math>
			is Hall's universal group ''U''.

			Indeed, ''U'' then contains a [[symmetric group]] of arbitrarily large order, and any
			group admits a monomorphism to a [[symmetric group\|group of permutations]], as explained above.
			Let ''G'' be a finite group admitting two embeddings to ''U''.
			Since ''U'' is a direct limit and ''G'' is finite, the
			images of these two embeddings belong to
			<math>\Gamma_i \subset U </math>. The group
			<math>\Gamma_{i+1}= S_{\Gamma_i}</math> acts on <math>\Gamma_i</math>
			by permutations, and conjugates all possible embeddings
			<math>G \hookrightarrow U</math>.

			==References==

			<references />

			[[Category:Infinite group theory]]
			[[Category:Permutation groups]]

en>ChrisGualtieri: /* S-duality */Typo fixing, typos fixed: etc → etc. using AWB

2012-07-24T03:16:27Z

S-duality: Typo fixing, typos fixed: etc → etc. using AWB

New page

The author is known as Araceli Gulledge. I've always cherished residing in Idaho. The thing she adores most is to perform handball but she can't make it her occupation. Bookkeeping has been his day job for a while.<br><br>My blog; [http://Www.Creativecats.com/UserProfile/tabid/331/userId/19522/Default.aspx Www.Creativecats.com]

String duality - Revision history

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q780099

en>ChrisGualtieri: /* S-duality */Typo fixing, typos fixed: etc → etc. using AWB