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{{Short description|Sporadic simple group}}
{{Short description|Sporadic simple group}}
{{About|the largest of the sporadic finite simple groups|the kind of infinite group known as a Tarski monster group| Tarski monster group}}
{{About|the largest of the sporadic finite simple groups|the kind of infinite group known as a Tarski monster group| Tarski monster group}}
{{Group theory sidebar |Finite}}
{{Use shortened footnotes|date=May 2021}}
{{Use shortened footnotes|date=May 2021}}


In the area of [[abstract algebra]] known as [[group theory]], the '''monster group''' M (also known as the '''Fischer–Griess monster''', or the '''friendly giant''') is the largest [[sporadic simple group]], having [[Order (group theory)|order]]
In the area of [[abstract algebra]] known as [[group theory]], the '''monster group''' M (also known as the '''Fischer–Griess monster''', or the '''friendly giant''') is the largest [[sporadic simple group]]; it has [[Order (group theory)|order]]
 
:   {{gaps|808|017|424|794|512|875|886|459|904|961|710|757|005|754|368|000|000|000}}
:   {{zwsp|808,|017,|424,|794,|512,|875,|886,|459,|904,|961,|710,|757,|005,|754,|368,|000,|000,|000}}
: = 2<sup>46</sup>{{·}}3<sup>20</sup>{{·}}5<sup>9</sup>{{·}}7<sup>6</sup>{{·}}11<sup>2</sup>{{·}}13<sup>3</sup>{{·}}17{{·}}19{{·}}23{{·}}29{{·}}31{{·}}41{{·}}47{{·}}59{{·}}71
: = 2<sup>46</sup>{{·}}3<sup>20</sup>{{·}}5<sup>9</sup>{{·}}7<sup>6</sup>{{·}}11<sup>2</sup>{{·}}13<sup>3</sup>{{·}}17{{·}}19{{·}}23{{·}}29{{·}}31{{·}}41{{·}}47{{·}}59{{·}}71
: ≈ 8{{e|53}}.
: = 32!{{·}}10!{{·}}4!<sup>2</sup>{{·}}2{{·}}7{{·}}13{{·}}41{{·}}47{{·}}59{{·}}71
: ≈ 8.08 x {{10^|53}}.
{{Group theory sidebar |Finite}}


The [[Finite group|finite]] [[simple group]]s have been completely [[Classification of finite simple groups|classified]]. Every such group belongs to one of 18 [[countably infinite]] families or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as [[subquotient]]s. [[Robert Griess]], who proved the existence of the monster in 1982, has called those 20 groups the ''happy family'', and the remaining six exceptions ''[[pariah group|pariahs]]''.
The [[Finite group|finite]] [[simple group]]s have been completely [[Classification of finite simple groups|classified]]. Every such group belongs to one of 18&nbsp;[[countably infinite]] families or is one of 26&nbsp;sporadic groups that do not follow such a systematic pattern. The monster group contains 20&nbsp;sporadic groups (including itself) as [[subquotient]]s. [[Robert Griess]], who proved the existence of the monster in 1982, has called those 20&nbsp;groups the ''happy family'', and the remaining six exceptions ''[[pariah group|pariahs]]''.


It is difficult to give a good constructive definition of the monster because of its complexity. [[Martin Gardner]] wrote a popular account of the monster group in his June 1980 [[Mathematical Games column]] in ''[[Scientific American]]''.{{sfn|Gardner|1980|pp=20–33}}
It is difficult to give a good constructive definition of the monster because of its complexity. [[Martin Gardner]] wrote a popular account of the monster group in his June 1980 [[Mathematical Games column]] in ''[[Scientific American]]''.{{sfn|Gardner|1980|pp=20–33}}


==History==
==History==
The monster was predicted by [[Bernd Fischer (mathematician)|Bernd Fischer]] (unpublished, about 1973) and [[Robert Griess]]{{sfn|Griess|1976|pp=113–118}} as a simple group containing a [[Double covering group|double cover]] of Fischer's [[baby monster group]] as a [[Centralizer and normalizer|centralizer]] of an [[Involution (group theory)|involution]].  Within a few months, the order of M was found by Griess using the [[Thompson order formula]], and Fischer, [[John Horton Conway|Conway]], Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the [[Thompson group (finite)|Thompson group]] and the [[Harada–Norton group]]. The [[Character theory|character table]] of the monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. Griess{{sfn|Griess|1982|pp=1–102}} constructed M as the [[automorphism group]] of the [[Griess algebra]], a 196,883-dimensional commutative [[nonassociative algebra]] over the real numbers; he first announced his construction in [[Ann Arbor]] on January 14, 1980. In his 1982 paper, he referred to the monster as the Friendly Giant, but this name has not been generally adopted. [[John Horton Conway|John Conway]]{{sfn|Conway|1985|pp=513–540}} and [[Jacques Tits]]{{sfn|Tits|1983|pp=105–122}}{{sfn|Tits|1984|pp=491–499}} subsequently simplified this construction.
The monster was predicted by [[Bernd Fischer (mathematician)|Bernd Fischer]] (unpublished, about 1973) and [[Robert Griess]]{{sfn|Griess|1975|pp=113–118}} as a simple group containing a [[Double covering group|double cover]] of Fischer's [[baby monster group]] as a [[Centralizer and normalizer|centralizer]] of an [[Involution (group theory)|involution]].  Within a few months, the order of {{math|M}} was found by Griess using the [[Thompson order formula]], and Fischer, [[John Horton Conway|Conway]], Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the [[Thompson group (finite)|Thompson group]] and the [[Harada–Norton group]]. The [[Character theory|character table]] of the monster, a {{nobr|{{math|194 × 194}} array,}} was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. Griess{{sfn|Griess|1982|pp=1–102}} constructed {{math|M}} as the [[automorphism group]] of the [[Griess algebra]], a {{nobr|{{gaps|196|883}} dimensional}} commutative [[nonassociative algebra]] over the real numbers; he first announced his construction in [[Ann Arbor]] on 14&nbsp;January 1980. In his 1982 paper, he referred to the monster as the "Friendly Giant", but this name has not been generally adopted. [[John Horton Conway|John Conway]]{{sfn|Conway|1985|pp=513–540}} and [[Jacques Tits]]{{sfn|Tits|1983|pp=105–122}}{{sfn|Tits|1984|pp=491–499}} subsequently simplified this construction.


Griess's construction showed that the monster exists. [[John G. Thompson|Thompson]]{{sfn|Thompson|1979|pp=340–346}} showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a 196,883-dimensional [[faithful representation]]. A proof of the existence of such a representation was announced by [[Simon P. Norton|Norton]],{{sfn|Norton|1985|pp=271–285}} though he never published the details. Griess, Meierfrankenfeld, and Segev gave the first complete published proof of the uniqueness of the monster (more precisely, they showed that a group with the same centralizers of involutions as the monster is isomorphic to the monster).{{sfn|Griess|Meierfrankenfeld|Segev|1989|pp=567–602}}
Griess's construction showed that the monster exists. [[John G. Thompson|Thompson]]{{sfn|Thompson|1979|pp=340–346}} showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a {{nobr|{{gaps|196|883}} dimensional}} [[faithful representation]]. A proof of the existence of such a representation was announced by [[Simon P. Norton|Norton]],{{sfn|Norton|1982|pp=271–285}} though he never published the details. Griess, Meierfrankenfeld, and Segev gave the first complete published proof of the uniqueness of the monster (more precisely, they showed that a group with the same centralizers of involutions as the monster is isomorphic to the monster).{{sfn|Griess|Meierfrankenfeld|Segev|1989|pp=567–602}}


The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: the [[Fischer group]] Fi<sub>24</sub>, the baby monster, and the [[Conway group]] Co<sub>1</sub>.
The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: The [[Fischer group]] Fi<sub>24</sub>, the [[Baby monster group|baby monster]], and the [[Conway group]] Co<sub>1</sub>.


The [[Schur multiplier]] and the [[outer automorphism group]] of the monster are both [[Trivial group|trivial]].
The [[Schur multiplier]] and the [[outer automorphism group]] of the monster are both [[Trivial group|trivial]].


==Representations==
==Representations==
The minimal degree of a [[Faithful representation|faithful]] complex representation is 47 × 59 × 71 = 196,883, hence is the product of the three largest [[prime divisor]]s of the order of M.
The minimal degree of a [[Faithful representation|faithful]] complex representation is {{nobr|&thinsp;{{math|47 × 59 × 71 {{=}} {{gaps|196|883}}}}&thinsp;,}} which is the product of the three largest [[prime divisor]]s of the order of {{math|M}}.
The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation.
The smallest faithful linear representation over any field has dimension {{gaps|196|882}} over the field with two elements, only one less than the dimension of the smallest faithful complex representation.


The smallest faithful permutation representation of the monster is on
The smallest faithful permutation representation of the monster is on
 
:&nbsp;&nbsp;&nbsp;{{gaps|97|239|461|142|009|186|000}}
:&nbsp;&nbsp;&nbsp;97,239,461,142,009,186,000
: = 2<sup>4</sup>{{·}}3<sup>7</sup>{{·}}5<sup>3</sup>{{·}}7<sup>4</sup>{{·}}11{{·}}13<sup>2</sup>{{·}}29{{·}}41{{·}}59{{·}}71 ≈ {{10^|20}}
: = 2<sup>4</sup>{{·}}3<sup>7</sup>{{·}}5<sup>3</sup>{{·}}7<sup>4</sup>{{·}}11{{·}}13<sup>2</sup>{{·}}29{{·}}41{{·}}59{{·}}71 ≈ 10<sup>20</sup>
 
points.
points.


The monster can be realized as a [[Galois group]] over the [[rational number]]s,{{sfn|Thompson|1984|p=443}} and as a [[Hurwitz group]].{{sfn|Wilson|2001|pp=367–374}}
The monster can be realized as a [[Galois group]] over the [[rational number]]s,{{sfn|Thompson|1984|p=443}} and as a [[Hurwitz group]].{{sfn|Wilson|2001|pp=367–374}}


The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A<sub>100</sub> and SL<sub>20</sub>(2) are far larger but easy to calculate with as they have "small" permutation or linear representations. [[Alternating group]]s, such as A<sub>100</sub>, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of [[group of Lie type|Lie type]], such as SL<sub>20</sub>(2), have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension 4370).
The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A<sub>100</sub> and SL<sub>20</sub>(2) are far larger but easy to calculate with as they have "small" permutation or linear representations. [[Alternating group]]s, such as A<sub>100</sub>, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of [[group of Lie type|Lie type]], such as SL<sub>20</sub>(2), have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the [[baby monster group|baby monster]], with a representation of dimension {{gaps|4|370}}).


=== Computer construction ===
=== Computer construction ===
Martin Seysen (2022) has implemented a fast [[Python (programming language)|Python]] package named [https://mmgroup.readthedocs.io/ mmgroup], which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by [[Robert Arnott Wilson|Robert A. Wilson]] in 2013.<ref>{{cite web |url=https://mmgroup.readthedocs.io/en/latest/api.html |title=The mmgroup API reference |last=Seysen |first=Martin |access-date=31 July 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2203.04223 |title=A fast implementation of the Monster group |class=math.GR |date=8 Mar 2022}}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2002.10921 |title=A computer-friendly construction of the monster |class=math.GR |date=13 May 2020}}</ref><ref>{{cite arXiv |last=Wilson |first=Robert A. |author-link=Robert A. Wilson (mathematician)|eprint=1310.5016 |title=The Monster and black-box groups |class=math.GR |date=18 Oct 2013}}</ref> The mmgroup software package has been used to find two new maximal subgroups of the monster group.{{sfn|Dietrich|Lee|Popiel|2025|}}
Martin Seysen (2022) implemented a fast [[Python (programming language)|Python]] package named [https://mmgroup.readthedocs.io/ mmgroup], which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by [[Robert Arnott Wilson|Robert A. Wilson]] in 2013.<ref>{{cite web |last=Seysen |first=Martin |title=The mmgroup API reference |url=https://mmgroup.readthedocs.io/en/latest/api.html |via=mmgroup.readthedocs.io |access-date=31 July 2022 }}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |date=8 Mar 2022 |title=A fast implementation of the Monster group |eprint=2203.04223 |class=math.GR }}</ref><ref>{{cite arXiv |last=Seysen |first=Martin |author-link= |eprint=2002.10921 |title=A computer-friendly construction of the monster |class=math.GR |date=13 May 2020}}</ref><ref>{{cite arXiv |last=Wilson |first=Robert A. |author-link=Robert A. Wilson (mathematician) |date=18 Oct 2013 |title=The Monster and black-box groups |eprint=1310.5016 |class=math.GR }}</ref> The mmgroup software package has been used to find two new maximal subgroups of the monster group.{{sfn|Dietrich|Lee|Popiel|2025|}}


Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in [[GF(2)|the field of order 2]]) which together [[Generating set of a group|generate]] the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.{{sfn|Borcherds|2002|p=1076}}
Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in [[GF(2)|the field of order 2]]) which together [[Generating set of a group|generate]] the monster group by matrix multiplication; this is one dimension lower than the {{nobr|{{gaps|196|883}} dimensional}} representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.{{sfn|Borcherds|2002|p=1076}}


Wilson asserts that the best description of the monster is to say, "It is the [[automorphism group]] of the [[monster vertex algebra]]". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".{{sfn|Borcherds|2002|p=1077}}
Wilson asserts that the best description of the monster is to say, ''"It is the [[automorphism group]] of the [[monster vertex algebra]]"''. This is not much help however, because nobody has found a "really simple and natural construction of the ''monster vertex algebra''".{{sfn|Borcherds|2002|p=1077}}


Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let ''V'' be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup ''H'' (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup ''H'' chosen is 3<sup>1+12</sup>.2.Suz.2, where Suz is the [[Suzuki group (mathematics)|Suzuki group]]. Elements of the monster are stored as words in the elements of ''H'' and an extra generator ''T''. It is reasonably quick to calculate the action of one of these words on a vector in ''V''. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors ''u'' and ''v'' whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element ''g'' of the monster by finding the smallest ''i'' &gt; 0 such that ''g''<sup>''i''</sup>''u'' = ''u'' and ''g''<sup>''i''</sup>''v'' = ''v''. This and similar constructions (in different [[characteristic (algebra)|characteristics]]) were used to find some of the non-local maximal subgroups of the monster group.
Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let {{mvar|V}} be a 196,882 dimensional vector space over the field with 2&nbsp;elements. A large subgroup {{mvar|H}} (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup {{mvar|H}} chosen is 3<sup>1+12</sup>.2.Suz.2, where Suz is the [[Suzuki group (mathematics)|Suzuki group]]. Elements of the monster are stored as words in the elements of {{mvar|H}} and an extra generator {{mvar|T}}. It is reasonably quick to calculate the action of one of these words on a vector in {{mvar|V}}. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors {{mvar|u}} and {{mvar|v}} whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element {{mvar|g}} of the monster by finding the smallest {{nobr|&thinsp;{{math|''i'' &gt; 0}}&thinsp;}} such that {{nobr|&thinsp;{{math|''g''<sup>''i''</sup>''u'' {{=}} ''u''}}&thinsp;}} and {{nobr|&thinsp;{{math|''g''<sup>''i''</sup>''v'' {{=}} ''v''}}&thinsp;.}} This and similar constructions (in different [[characteristic (algebra)|characteristics]]) were used to find some of the non-local maximal subgroups of the monster group.


== Subquotients ==
== Subquotients ==
[[File:MonsterSporadicGroupGraph.svg|thumb|350px|Diagram of the 26 sporadic simple groups, showing subquotient relationships.]]
[[File:MonsterSporadicGroupGraph.svg|thumb|350px|Diagram of the 26&nbsp;sporadic simple groups, showing subquotient relationships.]]


The monster contains 20 of the 26 [[sporadic groups]] as subquotients. This diagram, based on one in the book ''Symmetry and the Monster'' by [[Mark Ronan]], shows how they fit together.{{sfn|Ronan|2006}} The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.
The monster contains 20 of the 26&nbsp;[[sporadic groups]] as subquotients. This diagram, based on one in the book ''Symmetry and the Monster'' by [[Mark Ronan]], shows how they fit together.{{sfn|Ronan|2006}} The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.


== Maximal subgroups ==
== Maximal subgroups ==


The monster has 46 conjugacy classes of maximal [[subgroups]].{{sfn|Dietrich|Lee|Popiel|2025|}} Non-abelian simple groups of some 60 [[isomorphism]] types are found as subgroups or as quotients of subgroups. The largest [[alternating group]] represented is A<sub>12</sub>.
The monster has 46&nbsp;conjugacy classes of maximal [[subgroups]].{{sfn|Dietrich|Lee|Popiel|2025}} Non-abelian simple groups of some 60&nbsp;[[isomorphism]] types are found as subgroups or as quotients of subgroups. The largest [[alternating group]] represented is A<sub>12</sub>.


The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple [[Socle of a group|socles]] of the form U<sub>3</sub>(4), L<sub>2</sub>(8), and L<sub>2</sub>(16).{{sfn|Wilson|2010|pp=393–403}}{{sfn|Norton|Wilson|2013|pp=943–962}}{{sfn|Wilson|2016|pp=355–364}} However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U<sub>3</sub>(4). The same authors had previously found a new maximal subgroup of the form L<sub>2</sub>(13) and confirmed that there are no maximal subgroups with socle L<sub>2</sub>(8) or L<sub>2</sub>(16), thus completing the classification in the literature.{{sfn|Dietrich|Lee|Popiel|2025|}}
The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple [[Socle of a group|socles]] of the form U<sub>3</sub>(4), L<sub>2</sub>(8), and L<sub>2</sub>(16).{{sfn|Wilson|2010|pp=393–403}}{{sfn|Norton|Wilson|2013|pp=943–962}}{{sfn|Wilson|2016|pp=355–364}} However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U<sub>3</sub>(4). The same authors had previously found a new maximal subgroup of the form L<sub>2</sub>(13) and confirmed that there are no maximal subgroups with socle L<sub>2</sub>(8) or L<sub>2</sub>(16), thus completing the classification in the literature.{{sfn|Dietrich|Lee|Popiel|2025|}}
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== Sources ==
== Sources ==
{{refbegin|30em}}
{{refbegin|25em|small=y}}
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* {{cite journal
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  | author-link = Richard Borcherds
| date = October 2002
  | journal = Notices of the American Mathematical Society
| title = What is ... the Monster?
  | date = October 2002 | volume = 49 | issue = 9
  | journal = [[Notices of the American Mathematical Society]]
  | volume = 49 | issue = 9
  | url = https://www.ams.org/notices/200209/what-is.pdf
  | url = https://www.ams.org/notices/200209/what-is.pdf
| via=ams.org
}}
}}
*{{cite web| title = The monster graph and McKay's observation
* {{cite web
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| date = 22 April 2009
| title = The monster graph and McKay's observation
  | website = neverendingbooks
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  | url = http://www.neverendingbooks.org/the-monster-graph-and-mckays-observation
| date = 22 April 2009
}}
}}
*{{Cite journal | title = A simple construction for the Fischer–Griess monster group
* {{cite journal
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  | last = Conway | first = J.H. | author-link = John Horton Conway
| author-link = John Horton Conway
| year = 1985
| title = A simple construction for the Fischer–Griess monster group
  | journal = [[Inventiones Mathematicae]]
  | journal = [[Inventiones Mathematicae]]
  | year = 1985 | volume = 79 | issue = 3 | pages = 513–540
  | volume = 79 | issue = 3 | pages = 513–540
  | bibcode = 1985InMat..79..513C | doi = 10.1007/BF01388521 | mr = 782233
  | bibcode = 1985InMat..79..513C | mr = 782233
| s2cid = 123340529  
| doi = 10.1007/BF01388521 | s2cid = 123340529
}}
}}
*{{Cite journal | title = Monstrous Moonshine
* {{cite journal
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  | author1-link = John Horton Conway
  | year  = 1979
  | author2-link = Simon P. Norton
  | title = Monstrous moonshine
  | journal = [[Bulletin of the London Mathematical Society]]
  | journal = [[Bulletin of the London Mathematical Society]]
  | year = 1979 | volume = 11 | issue = 3 | pages = 308–339
  | volume = 11 | issue = 3 | pages = 308–339
  | doi = 10.1112/blms/11.3.308  
  | doi = 10.1112/blms/11.3.308  
}}
}}
*{{cite journal |title = The maximal subgroups of the Monster
* {{cite journal
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| last1 = Dietrich | first1 = Heiko
| last2 = Lee      | first2 = Melissa
| last2 = Lee      | first2 = Melissa
| last3 = Popiel  | first3 = Tomasz
| last3 = Popiel  | first3 = Tomasz
| journal = Advances in Mathematics | date = 2025
| year = 2025
| volume = 469 | doi = 10.1016/j.aim.2025.110214 | arxiv = 2304.14646
| title = The maximal subgroups of the Monster
}} {{Bibcode|2023arXiv230414646D}}.
| journal = [[Advances in Mathematics]]
*{{cite arXiv| title = Arithmetic groups and the affine E8 Dynkin diagram
| volume = 469
| article-number=110214
| doi = 10.1016/j.aim.2025.110214 | arxiv = 2304.14646
| bibcode = 2023arXiv230414646D
}}
* {{cite arXiv
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| title = Arithmetic groups and the affine E8 Dynkin diagram
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| date = 2008
  | eprint = 0810.1465
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}}
}}
*{{Cite magazine| title = Mathematical games
* {{cite magazine
  | last = Gardner | first = Martin | year = 1980
  | last = Gardner | first = Martin | author-link = Martin Gardner
  | magazine = Scientific American
| date = June 1980
| title = The capture of the monster: A mathematical group with a ridiculous number of elements
| department = Mathematical games
  | magazine = [[Scientific American]]
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  | issn = 0036-8733 | jstor = 24966339
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| via  = scientificamerican.com
}}
}}
*{{Cite book| chapter = The structure of the monster simple group
* {{cite conference
  | last = Griess | first = Robert L. | year = 1976
  | last = Griess | first = Robert L.
  | title = Proceedings of the Conference on Finite Groups (Univ. Utah, 1975)
| year = 1975
| title = The structure of the monster simple group
  | book-title = Proceedings of the Conference on Finite Groups
| place = [[University of Utah]], Provo, UT
  | editor1-last = Scott | editor1-first = W. Richard
  | editor1-last = Scott | editor1-first = W. Richard
  | editor2-last = Gross | editor2-first = Fletcher
  | editor2-last = Gross | editor2-first = Fletcher
  | publisher = [[Academic Press]] | location = Boston, MA
  | publisher = [[Academic Press]]
| publication-date = 1976
| publication-place = Boston, MA
  | pages = 113–118
  | pages = 113–118
  | isbn = 978-012633650-4 | mr = 0399248
  | isbn = 978-012633650-4 | mr = 0399248
}}
}}
*{{Cite journal | title = The friendly giant
* {{cite journal
  | last = Griess | first = Robert L.
  | last = Griess | first = Robert L.
| year = 1982
| title = The friendly giant
  | journal = [[Inventiones Mathematicae]]
  | journal = [[Inventiones Mathematicae]]
  | year = 1982 | volume = 69 | issue = 1 | pages = 1–102
  | volume = 69 | issue = 1 | pages = 1–102
| bibcode = 1982InMat..69....1G | mr = 671653
| doi = 10.1007/BF01389186 | s2cid = 123597150
| hdl = 2027.42/46608 | hdl-access = free
  | url = https://deepblue.lib.umich.edu/bitstream/2027.42/46608/1/222_2005_Article_BF01389186.pdf
  | url = https://deepblue.lib.umich.edu/bitstream/2027.42/46608/1/222_2005_Article_BF01389186.pdf
  | bibcode = 1982InMat..69....1G | doi = 10.1007/BF01389186 | hdl = 2027.42/46608 | mr = 671653
  | via = U.Mich. (lib.umich.edu)
| s2cid = 123597150 | hdl-access = free
 
}}
}}
*{{Cite journal | title = A uniqueness proof for the Monster
* {{cite journal
  | last1 = Griess | first1 = Robert L.
  | last1 = Griess | first1 = Robert L.
  | last2 = Meierfrankenfeld | first2 = Ulrich
  | last2 = Meierfrankenfeld | first2 = Ulrich
  | last3 = Segev | first3 = Yoav
  | last3 = Segev | first3 = Yoav
| year = 1989
| title = A uniqueness proof for the Monster
  | journal = [[Annals of Mathematics]]
  | journal = [[Annals of Mathematics]]
  | year = 1989 | volume = 130 | issue = 3 | pages = 567–602
  | volume = 130 | issue = 3 | pages = 567–602
  | series = Second Series
  | series = Second Series
  | doi = 10.2307/1971455 | jstor = 1971455 | mr = 1025167
  | doi = 10.2307/1971455 | jstor = 1971455 | mr = 1025167
}}
}}
*{{Cite AV media| title = Life, Death and the Monster (John Conway)
* {{cite AV media
  | last = Haran | first = Brady
  | last = Haran | first = Brady
| year = 2014
| title = Life, death, and the Monster (John Conway)
  | publisher = [[Numberphile]]
  | publisher = [[Numberphile]]
  | at = 7:57
  | at = 7{{sup|m}}57{{sup|s}}
  | url = https://www.youtube.com/watch?v=xOCe5HUObD4 | via = [[YouTube]]
  | url = https://www.youtube.com/watch?v=xOCe5HUObD4
| date = 2014
| via = [[YouTube]]
}}
}}
*{{Cite arXiv| title = Sporadic and Exceptional
* {{cite arXiv
  | last1 = He | first1 = Yang-Hui
  | last1 = He   | first1 = Y.-H. | author1-link = Yang-Hui He
| author1-link = Yang-Hui He
  | last2 = McKay | first2 = John
  | last2 = McKay | first2 = John
| date = 25 May 2015
| title = Sporadic and Exceptional
  | class = math. AG
  | class = math. AG
| date = 25 May 2015
  | eprint = 1505.06742
  | eprint = 1505.06742
}}
}}
*{{Cite journal | title = A new maximal subgroup of the Monster
* {{cite journal
  | last1 = Holmes | first1 = Petra E.
  | last1 = Holmes | first1 = Petra E.
  | last2 = Wilson | first2 = Robert A.
  | last2 = Wilson | first2 = Robert A.
| year = 2002
| title = A new maximal subgroup of the Monster
  | journal = [[Journal of Algebra]]
  | journal = [[Journal of Algebra]]
  | year = 2002 | volume = 251 | issue = 1 | pages = 435–447
  | volume = 251 | issue = 1 | pages = 435–447
  | doi = 10.1006/jabr.2001.9037 | mr = 1900293
  | doi = 10.1006/jabr.2001.9037 | doi-access = free
  | doi-access = free
  | mr = 1900293
}}
}}
*{{Cite journal | title = PSL<sub>2</sub>(59) is a subgroup of the Monster
* {{cite journal
  | last1 = Holmes | first1 = Petra E.
  | last1 = Holmes | first1 = Petra E.
  | last2 = Wilson | first2 = Robert A.
  | last2 = Wilson | first2 = Robert A.
  | journal = Journal of the London Mathematical Society
| year = 2004
  | year = 2004 | volume = 69 | issue = 1 | pages = 141–152
| title = PSL<sub>2</sub>(59) is a subgroup of the Monster
  | journal = [[Journal of the London Mathematical Society]]
  | volume = 69 | issue = 1 | pages = 141–152
  | series = Second Series
  | series = Second Series
  | doi = 10.1112/S0024610703004915 | mr = 2025332
  | doi = 10.1112/S0024610703004915
| s2cid = 122913546 }}
| mr = 2025332 | s2cid = 122913546
*{{Cite journal | title = On subgroups of the Monster containing A<sub>5</sub>'s
}}
* {{cite journal
  | last1 = Holmes | first1 = Petra E.
  | last1 = Holmes | first1 = Petra E.
  | last2 = Wilson | first2 = Robert A.
  | last2 = Wilson | first2 = Robert A.
| year = 2008
| title = On subgroups of the Monster containing A<sub>5</sub>'s
  | journal = [[Journal of Algebra]]
  | journal = [[Journal of Algebra]]
  | year = 2008 | volume = 319 | issue = 7 | pages = 2653–2667
  | volume = 319 | issue = 7 | pages = 2653–2667
  | doi = 10.1016/j.jalgebra.2003.11.014 | mr = 2397402
  | doi = 10.1016/j.jalgebra.2003.11.014 | doi-access = free
  | doi-access = free
  | mr = 2397402
}}
}}
*{{Cite news| title = Simon Norton obituary
* {{cite news
  | last = Masters | first = Alexander
  | last = Masters | first = Alexander
  | newspaper = The Guardian
| date = 22 February 2019
| title = Simon Norton
| department = Obituary
  | newspaper = [[The Guardian]]
  | url = https://www.theguardian.com/education/2019/feb/22/simon-norton-obituary
  | url = https://www.theguardian.com/education/2019/feb/22/simon-norton-obituary
| date = 22 February 2019
}}
}}
*{{Cite book| chapter = The uniqueness of the Fischer–Griess Monster
* {{cite conference
  | last = Norton | first = Simon P. | year = 1985
  | last = Norton | first = Simon P.
  | title = Finite groups—coming of age (Montreal, Que., 1982)
| year = 1982
  | publisher = [[American Mathematical Society]] | location = Providence RI
  | title = The uniqueness of the Fischer–Griess Monster
  | volume = 45 | series = Contemp. Math.
| conference = Finite groups – coming of age
  | pages = 271–285
| place = Montreal, QC
  | doi = 10.1090/conm/045/822242 | isbn = 978-082185047-3 | mr = 822242
  | publisher = [[American Mathematical Society]]
| publication-place = Providence, RI
  | publication-date = 1985
| series = Contemp. Math.
  | volume = 45 | pages = 271–285
  | doi = 10.1090/conm/045/822242
| isbn = 978-082185047-3 | mr = 822242
}}
}}
*{{Cite journal | title = A correction to the 41-structure of the Monster, a construction of a new maximal subgroup L2(41) and a new Moonshine phenomenon
* {{cite journal
  | last1 = Norton | first1 = Simon P.
  | last1 = Norton | first1 = Simon P.
  | last2 = Wilson | first2 = Robert A.
  | last2 = Wilson | first2 = Robert A.
  | journal = Journal of the London Mathematical Society
| year = 2013
| year = 2013 | volume = 87 | issue = 3 | pages = 943–962
| title = A correction to the 41&nbsp;structure of the Monster, a construction of a new maximal subgroup L2(41) and a new Moonshine phenomenon
  | journal = [[Journal of the London Mathematical Society]]
  | series = Second Series
  | series = Second Series
| volume = 87 | issue = 3 | pages = 943–962
| doi = 10.1112/jlms/jds078
| s2cid = 7075719
  | url = http://www.maths.qmul.ac.uk/~raw/pubs_files/ML241sub.pdf
  | url = http://www.maths.qmul.ac.uk/~raw/pubs_files/ML241sub.pdf
  | doi = 10.1112/jlms/jds078
  | via=Q.Mary.U.London
| s2cid = 7075719 }}
}}
*{{Cite book| title = Curiosities: Pursuing the Monster
* {{cite book
  | last = Roberts | first = Siobhan | year = 2013
  | last = Roberts | first = Siobhan
  | publisher = Institute for Advanced Study
| year = 2013
| title = Curiosities: Pursuing the Monster
  | publisher = [[Institute for Advanced Study]]
| place = Princeton, NJ
  | url = https://www.ias.edu/ideas/2013/roberts-monster
  | url = https://www.ias.edu/ideas/2013/roberts-monster
}}
}}
*{{Cite book| title = Symmetry and the Monster
* {{cite book
  | last = Ronan | first = M. | year = 2006
  | last = Ronan | first = M. | author-link = Mark Ronan
  | author-link = Mark Ronan
| year = 2006
  | title = Symmetry and the Monster
  | publisher = Oxford University Press
  | publisher = Oxford University Press
  | isbn = 019280722-6
  | isbn = 019280722-6
}}
}}
*{{Cite journal | title = Uniqueness of the Fischer-Griess monster
* {{cite journal
  | last = Thompson | first = John G.
  | last = Thompson | first = John G. | author-link = John G. Thompson
| author-link = John G. Thompson
| year = 1979
  | journal = The Bulletin of the London Mathematical Society
| title = Uniqueness of the Fischer-Griess monster
  | year = 1979 | volume = 11 | issue = 3 | pages = 340–346
  | journal = [[Bulletin of the London Mathematical Society]]
  | volume = 11 | issue = 3 | pages = 340–346
  | doi = 10.1112/blms/11.3.340 | mr = 554400
  | doi = 10.1112/blms/11.3.340 | mr = 554400
}}
}}
*{{Cite journal | title = Some finite groups which appear as Gal ''L''/''K'', where ''K'' ⊆ Q(μ<sub>n</sub>)
* {{cite journal
| last = Thompson | first = John G.
| last = Thompson | first = John G. | author-link = John G. Thompson
| author-link = John G. Thompson
| year = 1984
  | journal = Journal of Algebra
| title = Some finite groups which appear as Gal ''L''/''K'', where ''K'' ⊆ Q(μ<sub>n</sub>)
  | year = 1984 | volume = 89 | issue = 2 | pages = 437–499
  | journal = [[Journal of Algebra]]
  | doi = 10.1016/0021-8693(84)90228-X | mr = 751155
  | volume = 89 | issue = 2 | pages = 437–499
  | doi-access = free
  | doi = 10.1016/0021-8693(84)90228-X | doi-access = free
  | mr = 751155
}}
}}
*{{Cite journal | title = Le Monstre (d'après R. Griess, B. Fischer et al.)
* {{cite journal
| last = Tits | first = Jacques
| last = Tits | first = Jacques | author-link = Jacques Tits
| author-link = Jacques Tits
| year = 1983
| title = Le Monstre (d'après R.&nbsp;Griess, B.&nbsp;Fischer ''et al''.)
  | journal = [[Astérisque]]
  | journal = [[Astérisque]]
  | year = 1983 | issue = 121 | pages = 105–122
  | issue = 121 | pages = 105–122
| mr = 768956 | zbl = 0548.20010
  | url = http://www.numdam.org/item?id=SB_1983-1984__26__105_0
  | url = http://www.numdam.org/item?id=SB_1983-1984__26__105_0
| mr = 768956 | zbl = 0548.20010
}}
}}
*{{Cite journal | title = On R. Griess' "friendly giant"
* {{cite journal
  | last = Tits | first = Jacques
  | last = Tits | first = Jacques | author-link = Jacques Tits
| author-link = Jacques Tits
| year = 1984
| title = On R. Griess' "friendly giant"
  | journal = [[Inventiones Mathematicae]]
  | journal = [[Inventiones Mathematicae]]
  | year = 1984 | volume = 78 | issue = 3 | pages = 491–499
  | volume = 78 | issue = 3 | pages = 491–499
  | bibcode = 1984InMat..78..491T | doi = 10.1007/BF01388446 | mr = 768989
  | bibcode = 1984InMat..78..491T | doi = 10.1007/BF01388446
| s2cid = 122379975  
| mr = 768989 | s2cid = 122379975  
}}
}}
*{{cite journal | title = The Monster is a Hurwitz group
* {{cite journal
  | last = Wilson | first = Robert A.
  | last = Wilson | first = R.A. | author-link = Robert Arnott Wilson
| author-link = Robert Arnott Wilson
| year = 2001
  | journal = Journal of Group Theory
| title = The Monster is a Hurwitz group
  | year = 2001 | volume = 4 | issue = 4 | pages = 367–374
  | journal = [[Journal of Group Theory]]
  | volume = 4 | issue = 4 | pages = 367–374
| doi = 10.1515/jgth.2001.027 | mr = 1859175
  | url = http://web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps
  | url = http://web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps
| url-access = subscription |via=mat.bham.ac.uk/
  | archive-url = https://web.archive.org/web/20120305071856/http://web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps
  | archive-url = https://web.archive.org/web/20120305071856/http://web.mat.bham.ac.uk/R.A.Wilson/pubs/MHurwitz.ps
  | archive-date = 2012-03-05
  | archive-date = 2012-03-05  
| doi = 10.1515/jgth.2001.027 | mr = 1859175
| url-access = subscription
  }}
  }}
*{{Cite book| chapter = New computations in the Monster
* {{cite book
  | last = Wilson | first = Robert A. | year = 2010
  | last = Wilson | first = R.A. | author-link = Robert Arnott Wilson
  | title = Moonshine: the first quarter century and beyond
| year = 2010
| chapter = New computations in the Monster
  | title = Moonshine: The first quarter century and beyond
  | publisher = [[Cambridge University Press]]
  | publisher = [[Cambridge University Press]]
  | volume = 372 | series = London Math. Soc. Lecture Note Ser.
  | volume = 372 | series = [[London Mathematical Society|London Math. Soc.]] Lecture Note Ser.
  | pages = 393–403
  | pages = 393–403
  | isbn = 978-052110664-1 | mr = 2681789
  | isbn = 978-052110664-1 | mr = 2681789
}}
}}
*{{cite journal | title = Is the Suzuki group Sz(8) a subgroup of the Monster?
* {{cite journal
| last = Wilson | first = Robert A.
| last = Wilson | first = R.A.  | author-link = Robert Arnott Wilson
  | journal = Bulletin of the London Mathematical Society
| year = 2016
  | year = 2016 | volume = 48 | issue = 2 | pages = 355–364
| title = Is the Suzuki group Sz(8) a subgroup of the Monster?
  | journal = [[Bulletin of the London Mathematical Society]]
  | volume = 48 | issue = 2 | pages = 355–364
| doi = 10.1112/blms/bdw012 | mr = 3483073
| s2cid = 123219818
  | url = https://qmro.qmul.ac.uk/xmlui/bitstream/123456789/12414/1/Wilson%20Is%20Sz%20%288%29%20a%20subgroup%202016%20Accepted.pdf
  | url = https://qmro.qmul.ac.uk/xmlui/bitstream/123456789/12414/1/Wilson%20Is%20Sz%20%288%29%20a%20subgroup%202016%20Accepted.pdf
  | doi = 10.1112/blms/bdw012 | mr = 3483073
  | via=Q.Mary.U.London / research online (qmro.qmul.ac.uk)
| s2cid = 123219818 }}
}}
{{refend}}
{{refend}}


==Further reading==
==Further reading==
{{refbegin|35em}}
{{refbegin|25em|small=y}}
*{{Cite book| title = Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups
* {{cite book
  | last1 = Conway | first1 = J. H.
  | last1 = Conway | first1 = J.H. | author1-link = John Horton Conway
  | last2 = Curtis | first2 = R. T.
  | last2 = Curtis | first2 = R.T.
  | last3 = Norton | first3 = S. P.
  | last3 = Norton | first3 = S.P. | author3-link = Simon P. Norton
  | last4 = Parker | first4 = R. A.
  | last4 = Parker | first4 = R.A. | author4-link = Richard A. Parker
  | last5 = Wilson | first5 = R. A.
  | last5 = Wilson | first5 = R.A. | author5-link = Robert Arnott Wilson
| author1-link = John Horton Conway
| author3-link = Simon P. Norton
| author4-link = Richard A. Parker
| author5-link = Robert Arnott Wilson
  | year = 1985
  | year = 1985
  | others = with computational assistance from J. G. Thackray
| title = Atlas of Finite Groups: Maximal subgroups and ordinary characters for simple groups
  | others = Thackray, J.G. (computational assistance)
  | publisher = Oxford University Press
  | publisher = Oxford University Press
  | isbn = 978-019853199-9
  | isbn = 978-019853199-9
| ref = none
}}
}}
*{{Cite journal | title = Mathematics of the Monster
* {{cite journal
  | last = Harada | first = Koichiro
  | last = Harada | first = Koichiro | author-link = Koichiro Harada
| author-link = Koichiro Harada
| year = 2001
| title = Mathematics of the Monster
  | journal = Sugaku Expositions
  | journal = Sugaku Expositions
  | year = 2001 | volume = 14 | issue = 1 | pages = 55–71
  | volume = 14 | issue = 1 | pages = 55–71
  | mr = 1690763
  | mr = 1690763
| ref = none
}}
}}
*{{Cite journal | title = A computer construction of the Monster using 2-local subgroups
* {{cite journal
  | last1 = Holmes | first1 = P. E.
  | last1 = Holmes | first1 = P.E.
  | last2 = Wilson | first2 = R. A.
  | last2 = Wilson | first2 = R.A. | author2-link = Robert Arnott Wilson
| author2-link = Robert Arnott Wilson
| year = 2003
  | journal = Journal of the London Mathematical Society
| title = A computer construction of the Monster using 2-local subgroups
  | year = 2003 | volume = 67 | issue = 2 | pages = 346–364
  | journal = [[Journal of the London Mathematical Society]]
  | volume = 67 | issue = 2 | pages = 346–364
  | doi = 10.1112/S0024610702003976
  | doi = 10.1112/S0024610702003976
  | s2cid = 102338377 | ref = none
  | s2cid = 102338377
}}
}}
*{{Cite journal | title = A classification of subgroups of the Monster isomorphic to S<sub>4</sub> and an application
* {{cite journal
  | last = Holmes | first = Petra E.
  | last = Holmes | first = Petra E.
| year = 2008
| title = A classification of subgroups of the Monster isomorphic to S<sub>4</sub> and an application
  | journal = [[Journal of Algebra]]
  | journal = [[Journal of Algebra]]
  | year = 2008 | volume = 319 | issue = 8 | pages = 3089–3099
  | volume = 319 | issue = 8 | pages = 3089–3099
  | doi = 10.1016/j.jalgebra.2004.01.031 | mr = 2408306
  | doi = 10.1016/j.jalgebra.2004.01.031 | doi-access = free
| doi-access = free
  | mr = 2408306
  | ref = none
}}
}}
*{{cite book| title = The Monster Group and Majorana Involutions
* {{cite book
  | last = Ivanov | first = A.A. | year = 2009
  | last = Ivanov | first = A.A.
| year = 2009
| title = The Monster group and Majorana involutions
  | publisher = Cambridge University Press
  | publisher = Cambridge University Press
  | volume = 176 | series = Cambridge tracts in mathematics
  | volume = 176 | series = Cambridge tracts in mathematics
  | doi = 10.1017/CBO9780511576812 | url = https://doi.org/10.1017/CBO9780511576812
  | doi = 10.1017/CBO9780511576812
  | isbn = 978-052188994-0
  | isbn = 978-052188994-0
  | ref = none
  | url = https://doi.org/10.1017/CBO9780511576812
}}
}}
*{{Cite book| chapter = Anatomy of the Monster. I
* {{cite conference
  | last = Norton | first = Simon P. | year = 1998
  | last = Norton | first = Simon P.
  | title = The atlas of finite groups: ten years on (Birmingham, 1995)
| year = 1995
  | title = Anatomy of the Monster.&nbsp;I
| conference = The Atlas of Finite Groups: Ten years on
| place = Birmingham, UK
  | publisher = [[Cambridge University Press]]
  | publisher = [[Cambridge University Press]]
  | volume = 249 | series = London Math. Soc. Lecture Note Ser.
| publication-date = 1998
  | volume = 249 | series = [[London Mathematical Society|London Math. Soc.]] Lecture Note Ser.
  | pages = 198–214
  | pages = 198–214
  | doi = 10.1017/CBO9780511565830.020 | isbn = 978-052157587-4 | mr = 1647423
  | doi = 10.1017/CBO9780511565830.020
| ref = none
| isbn = 978-052157587-4 | mr = 1647423
}}
}}
*{{Cite journal | title = Anatomy of the Monster. II
* {{cite journal
  | last1 = Norton | first1 = Simon P.
  | last1 = Norton | first1 = Simon P.
  | last2 = Wilson | first2 = Robert A.
  | last2 = Wilson | first2 = Robert A.
| year = 2002
| title = Anatomy of the Monster.&nbsp;II
  | journal = Proceedings of the London Mathematical Society
  | journal = Proceedings of the London Mathematical Society
  | year = 2002 | volume = 84 | issue = 3 | pages = 581–598
  | volume = 84 | issue = 3 | pages = 581–598
  | series = Third Series
  | series = Third Series
  | doi = 10.1112/S0024611502013357 | mr = 1888424
  | doi = 10.1112/S0024611502013357 | mr = 1888424
| ref = none
}}
}}
*{{Cite book| title = Finding Moonshine
* {{cite book
  | last = du Sautoy | first = Marcus | year = 2008
  | last = du&nbsp;Sautoy | first = M. | author-link = Marcus du Sautoy
  | author-link = Marcus du Sautoy
| year = 2008
  | title = Finding Moonshine
  | publisher = Fourth Estate
  | publisher = Fourth Estate
  | isbn = 978-000721461-7
  | isbn = 978-000721461-7
| ref = none
}} published in the US by HarperCollins as ''Symmetry'', {{isbn|978-006078940-4}}).
}} published in the US by HarperCollins as ''Symmetry'', {{isbn|978-006078940-4}}).
*{{Cite journal | title = Computer construction of the Monster
* {{cite journal
  | last1 = Wilson | first1 = R. A.
  | last1 = Wilson | first1 = R.A.
  | last2 = Walsh | first2 = P. G.
  | last2 = Walsh | first2 = P.G.
  | last3 = Parker | first3 = R. A.
  | last3 = Parker | first3 = R.A.
  | last4 = Linton | first4 = S. A.
  | last4 = Linton | first4 = S.A.
  | journal = Journal of Group Theory
| year = 1998
  | year = 1998 | volume = 1 | issue = 4 | pages = 307–337
| title = Computer construction of the Monster
  | journal = [[Journal of Group Theory]]
  | volume = 1 | issue = 4 | pages = 307–337
  | doi = 10.1515/jgth.1998.023 | ref = none
  | doi = 10.1515/jgth.1998.023 | ref = none
}}
}}
*{{Cite journal | title = Kashiwa Lectures on "New Approaches to the Monster"
* {{cite journal
  | last1 = McKay | first1 = John
  | last1 = McKay | first1 = John
  | last2 = He | first2 = Yang-Hui  
  | last2 = He   | first2 = Yang-Hui | author2-link = Yang-Hui He
| author2-link = Yang-Hui He
| journal = Notices of the ICCM
  | year = 2022
  | year = 2022
| volume = 10 | issue = 1 | pages = 71–88 | doi = 10.4310/ICCM.2022.v10.n1.a4 | arxiv = 2106.01162 | s2cid = 235293875 }}
| title = Kashiwa Lectures on "New approaches to the Monster"
| journal = Notices of the [[International Congress of Chinese Mathematicians|ICCM]]
| volume = 10 | issue = 1 | pages = 71–88
| doi = 10.4310/ICCM.2022.v10.n1.a4
| arxiv = 2106.01162 | s2cid = 235293875
}}
{{refend}}
{{refend}}


==External links==
==External links==
* [https://www.ams.org/notices/200209/what-is.pdf ''What is... The Monster?''] by [[Richard Borcherds|Richard E. Borcherds]], Notices of the [[American Mathematical Society]], October 2002 1077
* {{cite journal
* [http://mathworld.wolfram.com/MonsterGroup.html MathWorld: Monster Group]
|first=R.E. |last=Borcherds  |author-link=Richard Borcherds
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/ Atlas of Finite Group Representations: Monster group]
|date=October 2002x
* [http://www.scientificamerican.com/article/mathematical-games-1980-06/ Scientific American June 1980 Issue: The capture of the monster: a mathematical group with a ridiculous number of elements]
|title=What is ... the Monster?
|journal=[[Notices of the American Mathematical Society]]
|volume=1077
|url=https://www.ams.org/notices/200209/what-is.pdf
}}
* {{cite web
|title=Monster group
|website=MathWorld
|url=http://mathworld.wolfram.com/MonsterGroup.html
|via=mathworld.wolfram.com
}}
* {{cite web
|editor10-first=Robert A. |editor10-last=Wilson
|editor9-first=Peter      |editor9-last=Walsh
|editor8-first=Jonathan  |editor8-last=Tripp
|editor7-first=Ibrahim    |editor7-last=Suleiman
|editor6-first=Richard A. |editor6-last=Parker
|editor5-first=Simon      |editor5-last=Norton
|editor4-first=Simon J.  |editor4-last=Nickerson
|editor3-first=Steve      |editor3-last=Linton
|editor2-first=John N.    |editor2-last=Bray
|editor1-first=Rachel    |editor1-last=Abbott
|display-editors=6
|date=17 April 2024  |orig-date=4 May 1999
|title=Monster group
|department=Atlas of Finite Group Representations
|publisher=[[Queen Mary University of London]]
|series=School of Mathematical Sciences
|id=version&nbsp;2.0
|url=http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/
}}


{{Group navbox}}
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Latest revision as of 00:20, 30 October 2025

Template:Short description Script error: No such module "about". Template:Use shortened footnotes

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order

   Template:Gaps
= 246Template:·320Template:·59Template:·76Template:·112Template:·133Template:·17Template:·19Template:·23Template:·29Template:·31Template:·41Template:·47Template:·59Template:·71
= 32!Template:·10!Template:·4!2Template:·2Template:·7Template:·13Template:·41Template:·47Template:·59Template:·71
≈ 8.08 x Template:10^.

Template:Group theory sidebar

The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the happy family, and the remaining six exceptions pariahs.

It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in Scientific American.Template:Sfn

History

The monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert GriessTemplate:Sfn as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months, the order of Template:Math was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the monster, a Template:Nobr was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed. GriessTemplate:Sfn constructed Template:Math as the automorphism group of the Griess algebra, a Template:Nobr commutative nonassociative algebra over the real numbers; he first announced his construction in Ann Arbor on 14 January 1980. In his 1982 paper, he referred to the monster as the "Friendly Giant", but this name has not been generally adopted. John ConwayTemplate:Sfn and Jacques TitsTemplate:SfnTemplate:Sfn subsequently simplified this construction.

Griess's construction showed that the monster exists. ThompsonTemplate:Sfn showed that its uniqueness (as a simple group satisfying certain conditions coming from the classification of finite simple groups) would follow from the existence of a Template:Nobr faithful representation. A proof of the existence of such a representation was announced by Norton,Template:Sfn though he never published the details. Griess, Meierfrankenfeld, and Segev gave the first complete published proof of the uniqueness of the monster (more precisely, they showed that a group with the same centralizers of involutions as the monster is isomorphic to the monster).Template:Sfn

The monster was a culmination of the development of sporadic simple groups and can be built from any two of three subquotients: The Fischer group Fi24, the baby monster, and the Conway group Co1.

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

Representations

The minimal degree of a faithful complex representation is Template:Nobr which is the product of the three largest prime divisors of the order of Template:Math. The smallest faithful linear representation over any field has dimension Template:Gaps over the field with two elements, only one less than the dimension of the smallest faithful complex representation.

The smallest faithful permutation representation of the monster is on

   Template:Gaps
= 24Template:·37Template:·53Template:·74Template:·11Template:·132Template:·29Template:·41Template:·59Template:·71 ≈ Template:10^

points.

The monster can be realized as a Galois group over the rational numbers,Template:Sfn and as a Hurwitz group.Template:Sfn

The monster is unusual among simple groups in that there is no known easy way to represent its elements. This is not due so much to its size as to the absence of "small" representations. For example, the simple groups A100 and SL20(2) are far larger but easy to calculate with as they have "small" permutation or linear representations. Alternating groups, such as A100, have permutation representations that are "small" compared to the size of the group, and all finite simple groups of Lie type, such as SL20(2), have linear representations that are "small" compared to the size of the group. All sporadic groups other than the monster also have linear representations small enough that they are easy to work with on a computer (the next hardest case after the monster is the baby monster, with a representation of dimension Template:Gaps).

Computer construction

Martin Seysen (2022) implemented a fast Python package named mmgroup, which claims to be the first implementation of the monster group where arbitrary operations can effectively be performed. The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by Robert A. Wilson in 2013.[1][2][3][4] The mmgroup software package has been used to find two new maximal subgroups of the monster group.Template:Sfn

Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the Template:Nobr representation in characteristic 0. Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.Template:Sfn

Wilson asserts that the best description of the monster is to say, "It is the automorphism group of the monster vertex algebra". This is not much help however, because nobody has found a "really simple and natural construction of the monster vertex algebra".Template:Sfn

Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work. Let Template:Mvar be a 196,882 dimensional vector space over the field with 2 elements. A large subgroup Template:Mvar (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup Template:Mvar chosen is 31+12.2.Suz.2, where Suz is the Suzuki group. Elements of the monster are stored as words in the elements of Template:Mvar and an extra generator Template:Mvar. It is reasonably quick to calculate the action of one of these words on a vector in Template:Mvar. Using this action, it is possible to perform calculations (such as the order of an element of the monster). Wilson has exhibited vectors Template:Mvar and Template:Mvar whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element Template:Mvar of the monster by finding the smallest Template:Nobr such that Template:Nobr and Template:Nobr This and similar constructions (in different characteristics) were used to find some of the non-local maximal subgroups of the monster group.

Subquotients

File:MonsterSporadicGroupGraph.svg
Diagram of the 26 sporadic simple groups, showing subquotient relationships.

The monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together.Template:Sfn The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.

Maximal subgroups

The monster has 46 conjugacy classes of maximal subgroups.Template:Sfn Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A12.

The 46 classes of maximal subgroups of the monster are given by the following table. Previous unpublished work of Wilson et. al had purported to rule out any almost simple subgroups with non-abelian simple socles of the form U3(4), L2(8), and L2(16).Template:SfnTemplate:SfnTemplate:Sfn However, the latter was contradicted by Dietrich et al., who found a new maximal subgroup of the form U3(4). The same authors had previously found a new maximal subgroup of the form L2(13) and confirmed that there are no maximal subgroups with socle L2(8) or L2(16), thus completing the classification in the literature.Template:Sfn

Maximal subgroups of the Monster
No. Structure Order Comments
1 2 · B 8,309,562,962,452,852,382,355,161,088,000,000
= 242·313·56·72·11·13·17·19·23·31·47		 centralizer of an involution of class 2A; contains the normalizer (47:23) × 2 of a Sylow 47-subgroup
2 2Template:Su · Co1 139,511,839,126,336,328,171,520,000
= 246·39·54·72·11·13·23		 centralizer of an involution of class 2B
3 3 · Fi24 7,531,234,255,143,970,327,756,800
= 222·317·52·73·11·13·17·23·29		 normalizer of a subgroup of order 3 (class 3A); contains the normalizer ((29:14) × 3).2 of a Sylow 29-subgroup
4 22 · 2E6(2):S3 1,836,779,512,410,596,494,540,800
= 239·310·52·72·11·13·17·19		 normalizer of a Klein 4-group of type 2A2
5 210+16 · OTemplate:Su(2) 1,577,011,055,923,770,163,200
= 246·35·52·7·17·31
6 22+11+22.(S3 × M24) 50,472,333,605,150,392,320
= 246·34·5·7·11·23		 normalizer of a Klein 4-group; contains the normalizer (23:11) × S4 of a Sylow 23-subgroup
7 3Template:Su.2Suz.2 2,859,230,155,080,499,200
= 215·320·52·7·11·13		 normalizer of a subgroup of order 3 (class 3B)
8 25+10+20.(S3 × L5(2)) 2,061,452,360,684,666,880
= 246·33·5·7·31
9 S3 × Th 544,475,663,327,232,000
= 216·311·53·72·13·19·31		 normalizer of a subgroup of order 3 (class 3C); contains the normalizer (31:15) × S3 of a Sylow 31-subgroup
10 23+6+12+18.(L3(2) × 3S6) 199,495,389,743,677,440
= 246·34·5·7
11 38 · OTemplate:Su(3) · 23 133,214,132,225,341,440
= 211·320·5·7·13·41
12 (D10 × HN).2 5,460,618,240,000,000
= 216·36·57·7·11·19		 normalizer of a subgroup of order 5 (class 5A)
13 (32:2 × OTemplate:Su(3)).S4 2,139,341,679,820,800
= 216·315·52·7·13
14 32+5+10.(M11 × 2S4) 49,093,924,366,080
= 28·320·5·11
15 33+2+6+6:(L3(3) × SD16) 11,604,018,486,528
= 28·320·13
16 5Template:Su:2J2:4 378,000,000,000
= 210·33·59·7		 normalizer of a subgroup of order 5 (class 5B)
17 (7:3 × He):2 169,276,262,400
= 211·34·52·74·17		 normalizer of a subgroup of order 7 (class 7A)
18 (A5 × A12):2 28,740,096,000
= 212·36·53·7·11
19 53+3.(2 × L3(5)) 11,625,000,000
= 26·3·59·31
20 (A6 × A6 × A6).(2 × S4) 2,239,488,000
= 213·37·53
21 (A5 × U3(8):31):2 1,985,679,360
= 212·36·5·7·19		 contains the normalizer ((19:9) × A5):2 of a Sylow 19-subgroup
22 52+2+4:(S3 × GL2(5)) 1,125,000,000
= 26·32·59
23 (L3(2) × S4(4):2).2 658,022,400
= 213·33·52·7·17		 contains the normalizer ((17:8) × L3(2)).2 of a Sylow 17-subgroup
24 7Template:Su:(3 × 2S7) 508,243,680
= 25·33·5·76		 normalizer of a subgroup of order 7 (class 7B)
25 (52:4.22 × U3(5)).S3 302,400,000
= 29·33·55·7
26 (L2(11) × M12):2 125,452,800
= 29·34·52·112		 contains the normalizer (11:5 × M12):2 of a subgroup of order 11
27 (A7 × (A5 × A5):22):2 72,576,000
= 210·34·53·7
28 54:(3 × 2L2(25)):22 58,500,000
= 25·32·56·13
29 72+1+2:GL2(7) 33,882,912
= 25·32·76
30 M11 × A6.22 11,404,800
= 29·34·52·11
31 (S5 × S5 × S5):S3 10,368,000
= 210·34·53
32 (L2(11) × L2(11)):4 1,742,400
= 26·32·52·112
33 132:2L2(13).4 1,476,384
= 25·3·7·133
34 (72:(3 × 2A4) × L2(7)):2 1,185,408
= 27·33·73
35 (13:6 × L3(3)).2 876,096
= 26·34·132		 normalizer of a subgroup of order 13 (class 13A)
36 13Template:Su:(3 × 4S4) 632,736
= 25·32·133		 normalizer of a subgroup of order 13 (class 13B); normalizer of a Sylow 13-subgroup
37 U3(4):4 249,600
= 28·3·52·13		 Template:Sfn
38 L2(71) 178,920
= 23·32·5·7·71		 contains the normalizer 71:35 of a Sylow 71-subgroupTemplate:Sfn
39 112:(5 × 2A5) 72,600
= 23·3·52·112		 normalizer of a Sylow 11-subgroup.
40 L2(41) 34,440
= 23·3·5·7·41		 Norton and Wilson found a maximal subgroup of this form; due to a subtle error pointed out by Zavarnitsine some previous lists and papers stated that no such maximal subgroup existedTemplate:Sfn
41 L2(29):2 24,360
= 23·3·5·7·29		 Template:Sfn
42 72:SL2(7) 16,464
=24·3·73		 this was accidentally omitted from some previous lists of 7-local subgroups
43 L2(19):2 6,840
= 23·32·5·19		 Template:Sfn
44 L2(13):2 2,184
= 23·3·7·13		 Template:Sfn
45 59:29 1,711
= 29·59		 previously thought to be L2(59);Template:Sfn normalizer of a Sylow 59-subgroup
46 41:40 1,640
= 23·5·41		 normalizer of a Sylow 41-subgroup

Note that tables of maximal subgroups have often been found to contain subtle errors, and in particular at least two of the subgroups in this table were incorrectly omitted from some previous lists.

McKay's E8 observation

There are also connections between the monster and the extended Dynkin diagrams E~8 specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E8 observation.Template:SfnTemplate:SfnTemplate:Sfn This is then extended to a relation between the extended diagrams E~6,E~7,E~8 and the groups 3.Fi24Template:Prime, 2.B, and M, where these are (3/2/1-fold central extensions) of the Fischer group, baby monster group, and monster. These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the monster, and the order of the extension corresponds to the symmetries of the diagram. See ADE classification: trinities for further connections (of McKay correspondence type), including (for the monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known as Bring's curve.

Moonshine

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The monster group is one of two principal constituents in the monstrous moonshine conjecture by Conway and Norton,Template:Sfn which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992.

In this setting, the monster group is visible as the automorphism group of the monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the monster Lie algebra, a generalized Kac–Moody algebra.

Many mathematicians, including Conway, have seen the monster as a beautiful and still mysterious object.Template:Sfn Conway said of the monster group: "There's never been any kind of explanation of why it's there, and it's obviously not there just by coincidence. It's got too many intriguing properties for it all to be just an accident."Template:Sfn Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God."Template:Sfn

See also

  • Supersingular prime, the prime numbers that divide the order of the monster
  • Bimonster group, the wreath square of the monster group, which has a surprisingly simple presentation

Citations

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Sources

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Further reading

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External links

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