Thompson groups - Revision history

en>R Smalen: /* Amenability */The quote that amenability was conjectured by Goeghegan is very misleading. Thompson constructed this group to prove von Neumann conjecture and gave a sequence of lectures on this topic in late fifties and early sixties.

2014-11-20T21:18:49Z

Amenability: The quote that amenability was conjectured by Goeghegan is very misleading. Thompson constructed this group to prove von Neumann conjecture and gave a sequence of lectures on this topic in late fifties and early sixties.

← Older revision		Revision as of 23:18, 20 November 2014
Line 1:		Line 1:
	~~{{for\|the unrelated infinite simple groups found by Richard Thompson\|Thompson groups}}~~		Eusebio Stanfill is what's indicated on my birth document although it is in no way the name on particular birth certificate. Vermont could be where my home may. Software making has been my holiday weekend job for a if. To prepare is the only activity my wife doesn't approve of. You can acquire my website here: http://prometeu.net<br><br>Check out my web-site [http://prometeu.net clash of clans hack tool android]
	~~{{Group theory sidebar}}~~

	~~In [[group theory]], the~~ '~~''Thompson group''' ''Th'', found by {{harvs\|txt \|authorlink=John G. Thompson \|first=John G. \|last=Thompson \|year=1976}} and constructed by {{harvtxt\|Smith\|1976}},~~ is ~~a [[sporadic group\|sporadic]] [[simple group]] of [[order (group theory)\|order]]~~

	~~:   2<sup>15</sup>{{·}}3<sup>10</sup>{{·}}5<sup>3</sup>{{·}}7<sup>2</sup>{{·}}13{{·}}19{{·}}31~~
	~~: = 90745943887872000~~
	~~: ≈ 9{{·}}10<sup>16</sup>.~~

	~~Thompson and Smith constructed the Thompson group as the group of automorphisms of a certain lattice~~ in the ~~248-dimensional Lie algebra of E<sub>8</sub>~~. ~~It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the [[Chevalley group]] E<sub>8</sub>(3)~~. ~~The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the [[Dempwolff group]] (which unlike the Thompson group is~~ a ~~subgroup of the compact Lie group E<sub>8</sub>)~~.

	~~The centralizer of an element of order 3 of type 3C in the [[Monster group]]~~ is ~~a product of the Thompson group and a group of order 3, as a result of which~~ the ~~Thompson group acts on a [[vertex operator algebra]] over the field with 3 elements. This vertex operator algebra contains the E<sub>8</sub> Lie algebra over '''F'~~'~~'<sub>3</sub>, giving the embedding~~ of ~~''Th'' into E<sub>8</sub>(3).~~

	~~The [[Schur multiplier]] and the [[outer automorphism group]] of the Thompson group are both trivial.~~

	~~The Thompson group contains the [[Dempwolff group]] as a maximal subgroup~~.

	~~{{harvtxt\|Linton\|1989}} found the 16 classes of maximal subgroups of the Thompson group, as follows~~:
	*<math> 2_+^{1+8}\cdot A_9</math>,
	*<math>2^5\cdot L_5(2)</math>,
	*<math>(3\times G_2(3)): 2</math>,
	*<math>(3^3\times 3_+^{1+2})\cdot 3_+^{1+2}: 2S_4</math>,
	*<math>3^2\cdot 3^7:2S_4</math>,
	*<math>(3\times 3^4:2\cdot A_6): 2</math>,
	*<math>5_+^{1+2}: 4S_4</math>,
	*<math>5^2:GL_2(5)</math>,
	*<math>7^2:(3\times 2S_4)</math>,
	*<math>31:15</math>,
	*[[³D₄\|<math>{}^3D_4(2): 3</math>]],
	*<math>U_3(8): 6</~~math>,~~
	*<math>L_2(19)</~~math>,~~
	*<math>L_3(3)</math>,
	*<math>M_{10}</math>,
	*<math>S_5</math>.

	~~==References==~~
	*{{Citation \| last1=Linton \| first1=Stephen A. \| title=The maximal subgroups of the Thompson group \| doi=10.1112/jlms/s2-39.1.79 \| mr=989921 \| year=1989 \| journal=Journal of the London Mathematical Society. Second Series \| issn=0024-6107 \| volume=39 \| issue=1 \| pages=79–88}}
	*{{Citation \| last1=Smith \| first1=P. E. ~~\| title=A simple subgroup of M? and E~~<~~sub~~>8<~~/sub~~>~~(3) \| doi=10.1112/blms/8.2.161 \| mr=0409630 \| year=1976 \| journal=The Bulletin of the London Mathematical Society \| issn=0024~~-~~6093 \| volume=8 \| issue=2 \| pages=161–165}}~~
	*{{Citation \| last1=Thompson \| first1=John G. \| author1-link=John G. Thompson \| title=A conjugacy theorem for E<sub>8</sub> \| doi=10.1016/0021-8693(76)90235-0 \| mr=0399193 \| year=1976 \| journal=[[Journal of Algebra]] \| issn=0021-8693 \| volume=38 \| issue=2 \| pages=525–530}}

	~~== External links ==~~
	* [http://~~mathworld~~.~~wolfram.com/ThompsonGroup.html MathWorld: Thompson group]~~
	* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/ Atlas of ~~Finite Group Representations: Thompson group]~~

	~~[[Category:Sporadic groups]~~]

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-30T22:14:31Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 00:14, 31 January 2014
Line 1:		Line 1:
	~~Gabrielle Straub~~ is ~~what~~ the ~~person can call me although it's~~ not the a ~~large number~~ of ~~feminine~~ of ~~names~~. ~~Fish keeping~~ is ~~what I might every week~~. ~~Managing people is my day job now~~. ~~My house is now in South Carolina~~. ~~Go to my web to find out more~~: ~~http~~://~~circuspartypanama~~.~~com~~<br><br>~~my blog~~ - [http://~~circuspartypanama~~.com ~~clash~~ of ~~clans cheat~~]		{{for\|the unrelated infinite simple groups found by Richard Thompson\|Thompson groups}}
			{{Group theory sidebar}}

			In [[group theory]], the '''Thompson group''' ''Th'', found by {{harvs\|txt \|authorlink=John G. Thompson \|first=John G. \|last=Thompson \|year=1976}} and constructed by {{harvtxt\|Smith\|1976}}, is a [[sporadic group\|sporadic]] [[simple group]] of [[order (group theory)\|order]]

			:   2<sup>15</sup>{{·}}3<sup>10</sup>{{·}}5<sup>3</sup>{{·}}7<sup>2</sup>{{·}}13{{·}}19{{·}}31
			: = 90745943887872000
			: ≈ 9{{·}}10<sup>16</sup>.

			Thompson and Smith constructed the Thompson group as the group of automorphisms of a certain lattice in the 248-dimensional Lie algebra of E<sub>8</sub>. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the [[Chevalley group]] E<sub>8</sub>(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the [[Dempwolff group]] (which unlike the Thompson group is a subgroup of the compact Lie group E<sub>8</sub>).

			The centralizer of an element of order 3 of type 3C in the [[Monster group]] is a product of the Thompson group and a group of order 3, as a result of which the Thompson group acts on a [[vertex operator algebra]] over the field with 3 elements. This vertex operator algebra contains the E<sub>8</sub> Lie algebra over '''F'''<sub>3</sub>, giving the embedding of ''Th'' into E<sub>8</sub>(3).

			The [[Schur multiplier]] and the [[outer automorphism group]] of the Thompson group are both trivial.

			The Thompson group contains the [[Dempwolff group]] as a maximal subgroup.

			{{harvtxt\|Linton\|1989}} found the 16 classes of maximal subgroups of the Thompson group, as follows:
			*<math> 2_+^{1+8}\cdot A_9</math>,
			*<math>2^5\cdot L_5(2)</math>,
			*<math>(3\times G_2(3)): 2</math>,
			*<math>(3^3\times 3_+^{1+2})\cdot 3_+^{1+2}: 2S_4</math>,
			*<math>3^2\cdot 3^7:2S_4</math>,
			*<math>(3\times 3^4:2\cdot A_6): 2</math>,
			*<math>5_+^{1+2}: 4S_4</math>,
			*<math>5^2:GL_2(5)</math>,
			*<math>7^2:(3\times 2S_4)</math>,
			*<math>31:15</math>,
			*[[³D₄\|<math>{}^3D_4(2): 3</math>]],
			*<math>U_3(8): 6</math>,
			*<math>L_2(19)</math>,
			*<math>L_3(3)</math>,
			*<math>M_{10}</math>,
			*<math>S_5</math>.

			==References==
			*{{Citation \| last1=Linton \| first1=Stephen A. \| title=The maximal subgroups of the Thompson group \| doi=10.1112/jlms/s2-39.1.79 \| mr=989921 \| year=1989 \| journal=Journal of the London Mathematical Society. Second Series \| issn=0024-6107 \| volume=39 \| issue=1 \| pages=79–88}}
			*{{Citation \| last1=Smith \| first1=P. E. \| title=A simple subgroup of M? and E<sub>8</sub>(3) \| doi=10.1112/blms/8.2.161 \| mr=0409630 \| year=1976 \| journal=The Bulletin of the London Mathematical Society \| issn=0024-6093 \| volume=8 \| issue=2 \| pages=161–165}}
			*{{Citation \| last1=Thompson \| first1=John G. \| author1-link=John G. Thompson \| title=A conjugacy theorem for E<sub>8</sub> \| doi=10.1016/0021-8693(76)90235-0 \| mr=0399193 \| year=1976 \| journal=[[Journal of Algebra]] \| issn=0021-8693 \| volume=38 \| issue=2 \| pages=525–530}}

			== External links ==
			* [http://mathworld.wolfram.com/ThompsonGroup.html MathWorld: Thompson group]
			* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/ Atlas of Finite Group Representations: Thompson group]

			[[Category:Sporadic groups]]

129.199.2.17: Undid revision 481709627 by 130.225.103.21 (talk)---it is indeed Richard Thompson, not JG (e.g. read the AMS notices article)

2012-04-18T20:51:31Z

Undid revision 481709627 by 130.225.103.21 (talk)---it is indeed Richard Thompson, not JG (e.g. read the AMS notices article)

New page

Gabrielle Straub is what the person can call me although it's not the a large number of feminine of names. Fish keeping is what I might every week. Managing people is my day job now. My house is now in South Carolina. Go to my web to find out more: http://circuspartypanama.com<br><br>my blog - [http://circuspartypanama.com clash of clans cheat]