|
|
| 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]]
| |
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
Check out my web-site clash of clans hack tool android