Jump to content

Thompson sporadic group: Difference between revisions

From Wikipedia, the free encyclopedia
Content deleted Content added
Unit721 (talk | contribs)
Maximal subgroups: Index column added
 
(14 intermediate revisions by 12 users not shown)
Line 1: Line 1:
{{About| the sporadic simple group| the three unusual infinite groups F, T and V found by Thompson | Thompson groups}}
{{Short description|Sporadic simple group}}
{{About| the sporadic simple group found by John G. Thompson | the three unusual infinite groups F, T and V found by Richard Thompson | Thompson groups}}
{{Group theory sidebar |Finite}}
{{Group theory sidebar |Finite}}


In the area of modern algebra known as [[group theory]], the '''Thompson group''' ''Th'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
In the area of modern algebra known as [[group theory]], the '''Thompson group''' ''Th'' is a [[sporadic simple group]] of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>15</sup>{{·}}3<sup>10</sup>{{·}}5<sup>3</sup>{{·}}7<sup>2</sup>{{·}}13{{·}}19{{·}}31
:&nbsp;&nbsp;&nbsp;90,745,943,887,872,000
: = 2<sup>15</sup>{{·}}3<sup>10</sup>{{·}}5<sup>3</sup>{{·}}7<sup>2</sup>{{·}}13{{·}}19{{·}}31
: = 90745943887872000
: ≈ 9{{e|16}}.
: ≈ 9{{e|16}}.


==History==
==History==
''Th'' is one of the 26 sporadic groups and was found by {{harvs|txt |authorlink=John G. Thompson |first=John G. |last=Thompson |year=1976}} and constructed by [[Geoff Smith (mathematician)|Geoff Smith]]. They constructed it as the [[automorphism group]] 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>).
''Th'' is one of the 26 sporadic groups and was found by {{harvs|txt |authorlink=John G. Thompson |first=John G. |last=Thompson |year=1976}} and constructed by {{harvtxt|Smith|1976}}. They constructed it as the [[automorphism group]] 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>).


==Representations==
==Representations==
Line 14: Line 15:
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 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 full normalizer of a 3C element in the Monster group is S<sub>3</sub> × Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the [[Baby monster group]], which therefore contains Th as a subgroup.
The [[Schur multiplier]] and the [[outer automorphism group]] of the Thompson group are both trivial.


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

==Generalized monstrous moonshine==


Conway and Norton suggested in their 1979 paper that [[monstrous moonshine]] is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
Conway and Norton suggested in their 1979 paper that [[monstrous moonshine]] is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups.
Line 23: Line 26:
:<math>T_{3C}(\tau) = \Big(j(3\tau)\Big)^{1/3} = \frac{1}{q}\,+\,248q^2\,+\,4124q^5\,+\,34752q^8\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots\,</math>
:<math>T_{3C}(\tau) = \Big(j(3\tau)\Big)^{1/3} = \frac{1}{q}\,+\,248q^2\,+\,4124q^5\,+\,34752q^8\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots\,</math>


and ''j''(''τ'') is the [[j-function]].
and ''j''(''τ'') is the [[j-invariant]].


==Maximal subgroups ==
==Maximal subgroups ==
{{harvtxt|Linton|1989}} found the 16 conjugacy classes of maximal subgroups of ''Th'' as follows:
{{harvtxt|Linton|1989}} found the 16 conjugacy classes of maximal subgroups of ''Th'' as follows:


{| class="wikitable"
* 2<sub>+</sub><sup>1+8</sup> · ''A''
|+ Maximal subgroups of ''Th''
* 2<sup>5</sup> · ''L''<sub>5</sub>(2) {{pad|.5em}} This is the [[Dempwolff group]]
|-
* (3 x ''G''<sub>2</sub>(3)) : 2
! No. !! Structure !! Order !! Index !! Comments
* (3<sup>3</sup> × 3<sub>+</sub><sup>1+2</sup>) · 3<sub>+</sub><sup>1+2</sup> : 2''S''<sub>4</sub>
|-
* 3<sup>2</sup> · 3<sup>7</sup> : 2''S''<sub>4</sub>
| 1||<sup>3</sup>''D''<sub>4</sub>(2) : 3 ||style="text-align:right;"|634,023,936 <br />=&nbsp;2<sup>12</sup>·3<sup>5</sup>·7<sup>2</sup>·13||align=right| 143,127,000<br />=&nbsp;2<sup>3</sup>·3<sup>5</sup>·5<sup>3</sup>·19·31 ||
* (3 × 3<sup>4</sup> : 2 · ''A''<sub>6</sub>) : 2
|-
* 5<sub>+</sub><sup>1+2</sup> : 4''S''<sub>4</sub>
| 2||2<sup>5 ·</sup>''L''<sub>5</sub>(2) ||style="text-align:right;"|319,979,520 <br />=&nbsp;2<sup>15</sup>·3<sup>2</sup>·5·7·31 ||align=right| 283,599,225<br />=&nbsp;3<sup>8</sup>·5<sup>2</sup>·7·13·19 || the [[Dempwolff group]]
* 5<sup>2</sup> : ''GL''<sub>2</sub>(5)
|-
* 7<sup>2</sup> : (3 × 2''S''<sub>4</sub>)
| 3||2{{su|a=l|b=+|p=1+8}}<sup> ·</sup>''A''<sub>9</sub> ||style="text-align:right;"| 92,897,280 <br />=&nbsp;2<sup>15</sup>·3<sup>4</sup>·5·7 ||align=right| 976,841,775<br />=&nbsp;3<sup>6</sup>·5<sup>2</sup>·7·13·19·31 || centralizer of involution
* 31 : 15
|-
*<sup>3</sup>''D''<sub>4</sub>(2) : 3
| 4||''U''<sub>3</sub>(8) : 6 ||style="text-align:right;"| 33,094,656 <br />=&nbsp;2<sup>10</sup>·3<sup>5</sup>·7·19 ||align=right| 2,742,012,000<br />=&nbsp;2<sup>5</sup>·3<sup>5</sup>·5<sup>3</sup>·7·13·31 ||
* ''U''<sub>3</sub>(8) : 6
|-
* ''L''<sub>2</sub>(19)
| 5||(3 x ''G''<sub>2</sub>(3)) : 2 ||style="text-align:right;"| 25,474,176 <br />=&nbsp;2<sup>7</sup>·3<sup>7</sup>·7·13 ||align=right| 3,562,272,000<br />=&nbsp;2<sup>8</sup>·3<sup>3</sup>·5<sup>3</sup>·7·19·31 || normalizer of a subgroup of order 3 (class 3A)
* ''L''<sub>3</sub>(3)
|-
* ''M''<sub>10</sub>
| 6||(3<sup>3</sup> × 3{{su|a=l|b=+|p=1+2}}) · 3{{su|a=l|b=+|p=1+2}} : 2''S''<sub>4</sub>||style="text-align:right;"| 944,784 <br />=&nbsp;2<sup>4</sup>·3<sup>10</sup> ||align=right| 96,049,408,000<br />=&nbsp;2<sup>11</sup>·5<sup>3</sup>·7<sup>2</sup>·13·19·31 || normalizer of a subgroup of order 3 (class 3B)
* ''S''<sub>5</sub>
|-
| 7||3<sup>2</sup> · 3<sup>7</sup> : 2''S''<sub>4</sub> ||style="text-align:right;"| 944,784 <br />=&nbsp;2<sup>4</sup>·3<sup>10</sup> ||align=right| 96,049,408,000<br />=&nbsp;2<sup>11</sup>·5<sup>3</sup>·7<sup>2</sup>·13·19·31 ||
|-
| 8||(3 × 3<sup>4</sup> : 2 · ''A''<sub>6</sub>) : 2 ||style="text-align:right;"| 349,920 <br />=&nbsp;2<sup>5</sup>·3<sup>7</sup>·5 ||align=right| 259,333,401,600<br />=&nbsp;2<sup>10</sup>·3<sup>3</sup>·5<sup>2</sup>·7<sup>2</sup>·13·19·31|| normalizer of a subgroup of order 3 (class 3C)
|-
| 9||5{{su|a=l|b=+|p=1+2}} : 4''S''<sub>4</sub> ||style="text-align:right;"| 12,000 <br />=&nbsp;2<sup>5</sup>·3·5<sup>3</sup> ||align=right| 7,562,161,990,656<br />=&nbsp;2<sup>10</sup>·3<sup>9</sup>·7<sup>2</sup>·13·19·31 || normalizer of a subgroup of order 5
|-
|10||5<sup>2</sup> : ''GL''<sub>2</sub>(5) ||style="text-align:right;"| 12,000 <br />=&nbsp;2<sup>5</sup>·3·5<sup>3</sup> ||align=right| 7,562,161,990,656<br />=&nbsp;2<sup>10</sup>·3<sup>9</sup>·7<sup>2</sup>·13·19·31 ||
|-
|11||7<sup>2</sup> : (3 × 2''S''<sub>4</sub>) ||style="text-align:right;"| 7,056 <br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·7<sup>2</sup> ||align=right| 12,860,819,712,000<br />=&nbsp;2<sup>11</sup>·3<sup>8</sup>·5<sup>3</sup>·13·19·31 ||
|-
|12||''L''<sub>2</sub>(19) : 2 ||style="text-align:right;"| 6,840 <br />=&nbsp;2<sup>3</sup>·3<sup>2</sup>·5·19 ||align=right| 13,266,950,860,800<br />=&nbsp;2<sup>12</sup>·3<sup>8</sup>·5<sup>2</sup>·7<sup>2</sup>·13·31 ||
|-
|13||''L''<sub>3</sub>(3) ||style="text-align:right;"| 5,616 <br />=&nbsp;2<sup>4</sup>·3<sup>3</sup>·13 ||align=right| 16,158,465,792,000<br />=&nbsp;2<sup>11</sup>·3<sup>7</sup>·5<sup>3</sup>·7<sup>2</sup>·19·31 ||
|-
|14||''M''<sub>10</sub> ||style="text-align:right;"| 720 <br />=&nbsp;2<sup>4</sup>·3<sup>2</sup>·5 ||align=right|126,036,033,177,600<br />=&nbsp;2<sup>11</sup>·3<sup>8</sup>·5<sup>2</sup>·7<sup>2</sup>·13·19·31||
|-
|15||31 : 15 ||style="text-align:right;"| 465 <br />=&nbsp;3·5·31 ||align=right|195,152,567,500,800<br />=&nbsp;2<sup>15</sup>·3<sup>9</sup>·5<sup>2</sup>·7<sup>2</sup>·13·19 ||
|-
|16||''S''<sub>5</sub> ||style="text-align:right;"| 120 <br />=&nbsp;2<sup>3</sup>·3·5 ||align=right|756,216,199,065,600<br />=&nbsp;2<sup>12</sup>·3<sup>9</sup>·5<sup>2</sup>·7<sup>2</sup>·13·19·31||
|}


==References==
==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=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 |series=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=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}}
*{{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| doi-access=free }}


== External links ==
== External links ==

Latest revision as of 09:44, 24 October 2024

In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order

   90,745,943,887,872,000
= 215 · 310 · 53 · 72 · 13 · 19 · 31
≈ 9×1016.

History

[edit]

Th is one of the 26 sporadic groups and was found by John G. Thompson (1976) and constructed by Smith (1976). They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8. 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 E8(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 E8).

Representations

[edit]

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 E8 Lie algebra over F3, giving the embedding of Th into E8(3).

The full normalizer of a 3C element in the Monster group is S3 × Th, so Th centralizes 3 involutions alongside the 3-cycle. These involutions are centralized by the Baby monster group, which therefore contains Th as a subgroup.

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

Generalized monstrous moonshine

[edit]

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Th, the relevant McKay-Thompson series is (OEISA007245),

and j(τ) is the j-invariant.

Maximal subgroups

[edit]

Linton (1989) found the 16 conjugacy classes of maximal subgroups of Th as follows:

Maximal subgroups of Th
No. Structure Order Index Comments
1 3D4(2) : 3 634,023,936
= 212·35·72·13
143,127,000
= 23·35·53·19·31
2 25 ·L5(2) 319,979,520
= 215·32·5·7·31
283,599,225
= 38·52·7·13·19
the Dempwolff group
3 21+8
+
·A9
92,897,280
= 215·34·5·7
976,841,775
= 36·52·7·13·19·31
centralizer of involution
4 U3(8) : 6 33,094,656
= 210·35·7·19
2,742,012,000
= 25·35·53·7·13·31
5 (3 x G2(3)) : 2 25,474,176
= 27·37·7·13
3,562,272,000
= 28·33·53·7·19·31
normalizer of a subgroup of order 3 (class 3A)
6 (33 × 31+2
+
) · 31+2
+
 : 2S4
944,784
= 24·310
96,049,408,000
= 211·53·72·13·19·31
normalizer of a subgroup of order 3 (class 3B)
7 32 · 37 : 2S4 944,784
= 24·310
96,049,408,000
= 211·53·72·13·19·31
8 (3 × 34 : 2 · A6) : 2 349,920
= 25·37·5
259,333,401,600
= 210·33·52·72·13·19·31
normalizer of a subgroup of order 3 (class 3C)
9 51+2
+
 : 4S4
12,000
= 25·3·53
7,562,161,990,656
= 210·39·72·13·19·31
normalizer of a subgroup of order 5
10 52 : GL2(5) 12,000
= 25·3·53
7,562,161,990,656
= 210·39·72·13·19·31
11 72 : (3 × 2S4) 7,056
= 24·32·72
12,860,819,712,000
= 211·38·53·13·19·31
12 L2(19) : 2 6,840
= 23·32·5·19
13,266,950,860,800
= 212·38·52·72·13·31
13 L3(3) 5,616
= 24·33·13
16,158,465,792,000
= 211·37·53·72·19·31
14 M10 720
= 24·32·5
126,036,033,177,600
= 211·38·52·72·13·19·31
15 31 : 15 465
= 3·5·31
195,152,567,500,800
= 215·39·52·72·13·19
16 S5 120
= 23·3·5
756,216,199,065,600
= 212·39·52·72·13·19·31

References

[edit]
  • Linton, Stephen A. (1989), "The maximal subgroups of the Thompson group", Journal of the London Mathematical Society, Second Series, 39 (1): 79–88, doi:10.1112/jlms/s2-39.1.79, ISSN 0024-6107, MR 0989921
  • Smith, P. E. (1976), "A simple subgroup of M? and E8(3)", The Bulletin of the London Mathematical Society, 8 (2): 161–165, doi:10.1112/blms/8.2.161, ISSN 0024-6093, MR 0409630
  • Thompson, John G. (1976), "A conjugacy theorem for E8", Journal of Algebra, 38 (2): 525–530, doi:10.1016/0021-8693(76)90235-0, ISSN 0021-8693, MR 0399193
[edit]