Sporadic group

{{Short description|Finite simple group type not classified as Lie, cyclic or alternating}} {{Group theory sidebar}}

In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the '''sporadic simple groups''', or the '''sporadic finite groups''', or just the '''sporadic groups'''.

A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families{{efn|content=The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups ²F₄(2^2''n''+1)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.}} plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type,<ref name=Conway>{{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985|loc=p. viii}}</ref> in which case there would be 27 sporadic groups.

The monster group, or ''friendly giant'', is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.<ref>{{harvtxt|Griess, Jr.|1998|p=146}}</ref>

== Names ==

Five of the sporadic groups were discovered by Émile Mathieu in the 1860s and the other twenty-one were found between 1965 (''J''₁) and 1975 (''J''₄). Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:<ref name=Conway /><ref>{{harvtxt|Gorenstein|Lyons|Solomon|1998|pp=262–302}}</ref><ref name=Ronan>{{harvtxt|Ronan|2006|pp=244–246}}</ref><br />

[[File:MonsterSporadicGroupGraph.svg|thumb|335px|The diagram shows the subquotient relations between the 26 '''sporadic groups'''. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between.<br>The generations of Robert Griess: File:EllipseSubqR.svg 1st, File:EllipseSubqG.svg 2nd, File:EllipseSubqB.svg 3rd, File:EllipseSubqW.svg Pariah]] * Mathieu groups ''M''₁₁, ''M''₁₂, ''M''₂₂, ''M''₂₃, ''M''₂₄ * Janko groups ''J''₁, ''J''₂ or ''HJ'', ''J''₃ or ''HJM'', ''J''₄ * Conway groups ''Co₁'', ''Co₂'', ''Co₃'' * Fischer groups ''Fi''₂₂, ''Fi''₂₃, ''Fi''₂₄&prime; or ''F''₃₊ * Higman-Sims group ''HS'' * McLaughlin group ''McL'' * Held group ''He'' or ''F''₇₊ or ''F''₇ * Rudvalis group ''Ru'' * Suzuki group ''Suz'' or ''F''_3&minus; * O'Nan group ''O'N'' (ON) * Harada-Norton group ''HN'' or ''F''₅₊ or ''F''₅ * Lyons group ''Ly'' * Thompson group ''Th'' or ''F''_3|3 or ''F''₃ * Baby Monster group ''B'' or ''F''₂₊ or ''F''₂ * Fischer-Griess Monster group ''M'' or ''F''₁

Various constructions for these groups were first compiled in {{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985}}, including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at {{harvtxt|Wilson|Parker|Nickerson|Bray|1999}}, updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or ''Brauer characters'' over fields of characteristic {{math|''p'' ≥ 0}} for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in {{harvtxt|Jansen|2005}}.

A further exception in the classification of finite simple groups is the Tits group {{math|''T''}}, which is sometimes considered of Lie type<ref>{{harvtxt|Howlett|Rylands|Taylor|2001|loc=p.429}} :"This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group)."</ref> or sporadic — it is almost but not strictly a group of Lie type<ref>{{harvtxt|Gorenstein|1979|loc=p.111}}</ref> — which is why in some sources the number of sporadic groups is given as 27, instead of 26.<ref name=Conway2>{{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985|loc=p.viii}}</ref><ref>{{harvtxt|Hartley|Hulpke|2010|loc=p.106}} :"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group {{nowrap|{{math|²''F''₄(2){{prime}}}}}} is counted also) which these infinite families do not include."</ref> In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both.<ref>{{harvtxt|Wilson|Parker|Nickerson|Bray|1999|loc=Sporadic groups & Exceptional groups of Lie type}}</ref>{{Citation needed |reason=In the literature, many different authors have included ''T'' as either being of Lie type, or sporadic, as well as both, and neither. Multiple sources referencing this would be very valuable, especially primary sourcing that mentions in detail arguments for both cases, if such a source exists. |date=November 2023}} The Tits group is the {{nowrap|(''n'' {{=}} 0)-member}} {{nowrap|{{math|²''F''₄(2){{prime}}}}}} of the infinite family of commutator groups {{nowrap|{{math|²''F''₄(2^2''n''+1){{prime}}}}}}; thus in a strict sense not sporadic, nor of Lie type. For {{nowrap|{{math|''n'' &gt; 0}}}} these finite simple groups coincide with the groups of Lie type {{nowrap|{{math|²''F''₄(2^2''n''+1)}},}} also known as Ree groups of type ²''F''₄.

The earliest use of the term ''sporadic group'' may be {{harvtxt|Burnside|1911|p=504}} where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.)

The diagram {{if mobile|above|at right}} is based on {{harvtxt|Ronan|2006|p=247}}. It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

== Organization ==

=== Happy Family ===

Of the 26 sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups (sections). These twenty have been called the ''happy family'' by Robert Griess, and can be organized into three generations.<ref>{{harvtxt|Griess, Jr.|1982|p=91}}</ref>{{efn|1={{harvtxt|Conway|Curtis|Norton|Parker|Wilson|1985|p=viii}} organizes the 26 sporadic groups in likeness: :"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."}}

==== First generation (5 groups): the Mathieu groups ====

{{main article|Mathieu groups}}

M_''n'' for ''n'' = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on ''n'' points. They are all subgroups of M₂₄, which is a permutation group on 24 points.<ref>{{harvtxt|Griess, Jr.|1998|pp=54–79}}</ref>

==== Second generation (7 groups): the Leech lattice ====

{{see also|Leech lattice|Conway groups}}

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:<ref>{{harvtxt|Griess, Jr.|1998|pp=104–145}}</ref>

* ''Co''₁ is the quotient of the automorphism group by its center {±1} * ''Co''₂ is the stabilizer of a type 2 (i.e., length 2) vector * ''Co''₃ is the stabilizer of a type 3 (i.e., length {{radic|6}}) vector * ''Suz'' is the group of automorphisms preserving a complex structure (modulo its center) * ''McL'' is the stabilizer of a type 2-2-3 triangle * ''HS'' is the stabilizer of a type 2-3-3 triangle * ''J''₂ is the group of automorphisms preserving a quaternionic structure (modulo its center).

==== Third generation (8 groups): other subgroups of the Monster ====

Consists of subgroups which are closely related to the Monster group ''M'':<ref>{{harvtxt|Griess, Jr.|1998|pp=146−150}}</ref>

* ''B'' or ''F''₂ has a double cover which is the centralizer of an element of order 2 in ''M'' * ''Fi''₂₄′ has a triple cover which is the centralizer of an element of order 3 in ''M'' (in conjugacy class "3A") * ''Fi''₂₃ is a subgroup of ''Fi''₂₄′ * ''Fi''₂₂ has a double cover which is a subgroup of ''Fi''₂₃ * The product of ''Th'' = ''F''₃ and a group of order 3 is the centralizer of an element of order 3 in ''M'' (in conjugacy class "3C") * The product of ''HN'' = ''F''₅ and a group of order 5 is the centralizer of an element of order 5 in ''M'' * The product of ''He'' = ''F''₇ and a group of order 7 is the centralizer of an element of order 7 in ''M''. * Finally, the Monster group itself is considered to be in this generation.

(This series continues further: the product of ''M''₁₂ and a group of order 11 is the centralizer of an element of order 11 in ''M''.)

The Tits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S₄ &times;²F₄(2)&prime; normalising a 2C² subgroup of ''B'', giving rise to a subgroup 2·S₄ &times;²F₄(2)′ normalising a certain Q₈ subgroup of the Monster. ²F₄(2)′ is also a subquotient of the Fischer group ''Fi''₂₂, and thus also of ''Fi''₂₃ and ''Fi''₂₄′, and of the Baby Monster ''B''. ²F₄(2)′ is also a subquotient of the (pariah) Rudvalis group ''Ru'', and has no involvements in sporadic simple groups except the ones already mentioned.

=== Pariahs ===

{{main article|Pariah group}}

The six exceptions are ''J''₁, ''J''₃, ''J''₄, ''O'N'', ''Ru'', and ''Ly'', sometimes known as the pariahs.<ref>{{harvtxt|Griess, Jr.|1982|pp=91−96}}</ref><ref>{{harvtxt|Griess, Jr.|1998|pp=146, 150−152}}</ref>

==Table of the sporadic group orders (with Tits group)==

{| class="wikitable sortable" |- ! class="sortable" style="vertical-align:bottom"| Group !style="vertical-align:bottom"| Discoverer !style="vertical-align:bottom"| <ref>{{harvtxt|Hiss|2003|loc=p. 172}} :Tabelle 2. Die Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)</ref><br /> Year !style="vertical-align:bottom"| {{Vertical text|Generation}} ! data-sort-type="number" style="vertical-align:bottom"|<ref name=Conway /><ref name=Ronan /><ref>{{harvtxt|Sloane|1996}}</ref><br /> Order !align=right style="vertical-align:bottom"|<ref>{{harvtxt|Jansen|2005|pp=122–123}}</ref><br /> Degree of minimal faithful Brauer character !style="vertical-align:top;"|<ref>{{harvtxt|Nickerson|Wilson|2011|loc=p. 365}}</ref><ref name=Wilsonetal>{{harvtxt|Wilson|Parker|Nickerson|Bray|1999}}</ref><br /><math>(a, b, ab)</math><br />{{Vertical text|Generators}} !style="vertical-align:bottom"|<ref name=Wilsonetal />{{efn|content=Here listed are semi-presentations from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed. }}<br /><math>\langle\langle a,b \mid o(z)\rangle\rangle</math><br />Semi-presentation |- | ''M'' or ''F''₁|| Fischer, Griess ||1973 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 808,017,424,794,512,875,886,459,904,961,710,<wbr>757,005,754,368,000,000,000<br />=&nbsp;2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31<wbr>·41·47·59·71<wbr> ≈&nbsp;8{{e|53}}|| align=center |196883||2A, 3B, 29||<math>o\bigl((ab)^{4}(ab^{2})^{2}\bigr) = 50</math> |- | ''B'' or ''F''₂|| Fischer ||1973 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 4,154,781,481,226,426,191,177,580,544,000,000<br />=&nbsp;2⁴¹·3¹³·5⁶·7²·11·13·17·19·23·31·47 ≈&nbsp;4{{e|33}}|| align=center |4371||2C, 3A, 55|| <math>o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 23</math> |- | ''Fi''₂₄ or ''F''₃₊|| Fischer ||1971 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 1,255,205,709,190,661,721,292,800<br />=&nbsp;2²¹·3¹⁶·5²·7³·11·13·17·23·29 ≈&nbsp;1{{e|24}}|| align=center |8671||2A, 3E, 29|| <math>o\bigl((ab)^3 b\bigr) = 33</math> |- | ''Fi''₂₃|| Fischer ||1971 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 4,089,470,473,293,004,800<br />=&nbsp;2¹⁸·3¹³·5²·7·11·13·17·23 ≈&nbsp;4{{e|18}}|| align=center |782||2B, 3D, 28||<math>o\bigl(a^{bb}(ab)^{14}\bigr) = 5</math> |- | ''Fi''₂₂|| Fischer ||1971 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 64,561,751,654,400<br />=&nbsp;2¹⁷·3⁹·5²·7·11·13 ≈&nbsp;6{{e|13}}|| align=center |78||2A, 13, 11||<math>o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 12</math> |- | ''Th'' or ''F''₃|| Thompson ||1976 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 90,745,943,887,872,000<br />=&nbsp;2¹⁵·3¹⁰·5³·7²·13·19·31 ≈&nbsp;9{{e|16}}|| align=center |248||2, 3A, 19||<math>o\bigl((ab)^{3}b\bigr) = 21</math> |- | ''Ly''|| Lyons ||1972 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 51,765,179,004,000,000<br />=&nbsp;2⁸·3⁷·5⁶·7·11·31·37·67 ≈&nbsp;5{{e|16}}|| align=center |2480||2, 5A, 14|| <math>o\bigl(ababab^2\bigr) = 67</math> |- | ''HN'' or ''F''₅|| Harada, Norton ||1976 | style="background-color:#aaccff;" | {{sort|3|3rd}} || align=right | 273,030,912,000,000<br />=&nbsp;2¹⁴·3⁶·5⁶·7·11·19 ≈&nbsp;3{{e|14}}|| align=center |133 ||2A, 3B, 22|| <math>o\bigl([a, b]\bigr) = 5</math> |- | ''Co''₁|| Conway ||1969 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 4,157,776,806,543,360,000<br />=&nbsp;2²¹·3⁹·5⁴·7²·11·13·23 ≈&nbsp;4{{e|18}}|| align=center |276||2B, 3C, 40||<math>o\bigl(ab(abab^{2})^{2}\bigr) = 42</math> |- | ''Co''₂|| Conway ||1969 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 42,305,421,312,000<br />=&nbsp;2¹⁸·3⁶·5³·7·11·23 ≈&nbsp;4{{e|13}}|| align=center |23||2A, 5A, 28||<math>o\bigl([a,b]\bigr) = 4</math> |- | ''Co''₃|| Conway ||1969 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 495,766,656,000<br />=&nbsp;2¹⁰·3⁷·5³·7·11·23 ≈&nbsp;5{{e|11}}|| align=center |23||2A, 7C, 17||<math>o\bigl((uvv)^{3}(uv)^{6}\bigr) = 5 </math>{{efn|content=Where <math>u = (b^{2}(b^{2})abb)^{3}</math> and <math>v = t(b^{2}(b^{2})t)^{2}</math> with <math>t = abab^{3}a^{2}</math>. }} |- | ''ON'' or ''O'N''|| O'Nan ||1976 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 460,815,505,920<br />=&nbsp;2⁹·3⁴·5·7³·11·19·31 ≈&nbsp;5{{e|11}}|| align=center |10944||2A, 4A, 11||<math>o\bigl(abab(b^{2}(b^{2})^{abab})^{5}\bigr) = 5</math> |- | ''Suz''|| Suzuki ||1969 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 448,345,497,600<br />=&nbsp;2¹³·3⁷·5²·7·11·13 ≈&nbsp;4{{e|11}}|| align=center |143||2B, 3B, 13|| <math>o\bigl([a, b]\bigr) = 15</math> |- | ''Ru''|| Rudvalis ||1972 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 145,926,144,000<br />=&nbsp;2¹⁴·3³·5³·7·13·29 ≈&nbsp;1{{e|11}}|| align=center |378||2B, 4A, 13||<math>o(abab^{2}) = 29</math> |- | ''He'' or ''F''₇|| Held ||1969 | style="background-color:#aaccff;" | {{sort|3|3rd}}|| align=right | 4,030,387,200<br />=&nbsp;2¹⁰·3³·5²·7³·17 ≈&nbsp;4{{e|9}} || align=center |51||2A, 7C, 17||<math>o\bigl(ab^{2}abab^{2}ab^{2}\bigr) = 10</math> |- | ''McL''|| McLaughlin ||1969 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 898,128,000<br />=&nbsp;2⁷·3⁶·5³·7·11 ≈&nbsp;9{{e|8}} || align=center |22||2A, 5A, 11|| <math>o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 7</math> |- | ''HS''|| Higman, Sims ||1967 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 44,352,000<br />=&nbsp;2⁹·3²·5³·7·11 ≈&nbsp;4{{e|7}} || align=center |22||2A, 5A, 11||<math>o(abab^{2}) = 15</math> |- | ''J''₄|| Janko ||1976 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 86,775,571,046,077,562,880<br />=&nbsp;2²¹·3³·5·7·11³·23·29·31·37·43 ≈&nbsp;9{{e|19}}|| align=center |1333||2A, 4A, 37|| <math>o\bigl(abab^2\bigr) = 10</math> |- | ''J''₃ or ''HJM''|| Janko ||1968 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 50,232,960<br />=&nbsp;2⁷·3⁵·5·17·19 ≈&nbsp;5{{e|7}} || align=center |85||2A, 3A, 19|| <math>o\bigl([a, b]\bigr) = 9</math> |- | ''J''₂ or ''HJ''|| Janko ||1968 | style="background-color:#80ff80;" | {{sort|2|2nd}}|| align=right | 604,800<br />=&nbsp;2⁷·3³·5²·7 ≈&nbsp;6{{e|5}} || align=center |14||2B, 3B, 7|| <math>o\bigl([a, b]\bigr) = 12</math> |- | ''J''₁|| Janko ||1965 | style="background-color:#ffffff;" | {{sort|4|Pariah}}|| align=right | 175,560<br />=&nbsp;2³·3·5·7·11·19 ≈&nbsp;2{{e|5}} || align=center |56||2, 3, 7|| <math>o\bigl(abab^2\bigr) = 19</math> |- | ''M''₂₄|| Mathieu ||1861 | style="background-color:#ffaaaa;" | {{sort|1|1st}}|| align=right | 244,823,040<br />=&nbsp;2¹⁰·3³·5·7·11·23 ≈&nbsp;2{{e|8}} || align=center |23||2B, 3A, 23|| <math>o\bigl(ab(abab^2)^2 ab^2\bigr) = 4</math> |- | ''M''₂₃|| Mathieu ||1861 | style="background-color:#ffaaaa;" | {{sort|1|1st}}|| align=right | 10,200,960<br />=&nbsp;2⁷·3²·5·7·11·23 ≈&nbsp;1{{e|7}} || align=center |22||2, 4, 23|| <math>o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 8</math> |- | ''M''₂₂|| Mathieu ||1861 | style="background-color:#ffaaaa;" | {{sort|1|1st}}|| align=right | 443,520<br />=&nbsp;2⁷·3²·5·7·11 ≈&nbsp;4{{e|5}} || align=center |21||2A, 4A, 11|| <math>o\bigl(abab^2\bigr) = 11</math> |- | ''M''₁₂|| Mathieu ||1861 | style="background-color:#ffaaaa;" | {{sort|1|1st}}|| align=right | 95,040<br />=&nbsp;2⁶·3³·5·11 ≈&nbsp;1{{e|5}} || align=center |11||2B, 3B, 11||<math>o\bigl([a,b]\bigr) = o\bigl(ababab^{2}\bigr) = 6</math> |- | ''M''₁₁|| Mathieu ||1861 | style="background-color:#ffaaaa;" | {{sort|1|1st}}|| align=right | 7,920<br />=&nbsp;2⁴·3²·5·11 ≈&nbsp;8{{e|3}} || align=center |10||2, 4, 11|| <math>o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 4</math> |- | ''T'' or <small>{{math|²''F''₄(2)′}}</small> || Tits ||1964 | style="background-color:#B9D9EB;" | {{sort|3|3rd}}|| align=right | 17,971,200<br />=&nbsp;2¹¹·3³·5²·13 ≈&nbsp;2{{e|7}} || align=center |104<ref>{{harvtxt|Lubeck|2001|p=2151}}</ref>||2A, 3, 13|| <math>o\bigl([a, b]\bigr) = 5</math> |}

== Notes ==

{{Notelist}}

== References ==

{{Reflist}}

=== Works cited ===

{{Refbegin}}

* {{Cite book |author-link=William Burnside |first=William |last=Burnside |year=1911 |title=Theory of groups of finite order |edition=2nd |publisher=Cambridge University Press |location=Cambridge |pages=xxiv, 1–512 |doi=10.1112/PLMS/S2-7.1.1 |isbn=0-486-49575-2 |hdl=2027/uc1.b4062919 |hdl-access=free |oclc=54407807 |mr=69818 |s2cid=117347785 |url=https://people.math.rochester.edu/faculty/doug/otherpapers/burnside1911.pdf}}

* {{Cite journal |author-link=John Horton Conway |last=Conway |first=J. H. |title=A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=61 |issue=2 |pages=398–400 |year=1968 |doi=10.1073/pnas.61.2.398 |doi-access=free |bibcode=1968PNAS...61..398C |zbl=0186.32401 |mr=237634 |pmid=16591697 |pmc=225171 |s2cid=29358882}}

* {{Cite book |last1=Conway |first1=J. H. |author1-link=John Horton Conway |last2=Curtis |first2=R. T. |last3=Norton |first3=S. P. |author3-link=Simon P. Norton |last4=Parker |first4=R. A. |author4-link=Richard A. Parker |last5=Wilson |first5=R. A. |author5-link=Robert Arnott Wilson |title=ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups |publisher=Clarendon Press |pages=xxxiii; 1–252 |year=1985 |location=Oxford |isbn=978-0-19-853199-9 |oclc=12106933 |mr=827219 |s2cid=117473588 |zbl=0568.20001}}

* {{Cite journal |last=Gorenstein |first=D. |author-link=Daniel Gorenstein |title=The classification of finite simple groups. I. Simple groups and local analysis |year=1979 |journal=Bulletin of the American Mathematical Society |series=New Series |publisher=American Mathematical Society |volume=1 |issue=1 |pages=43–199 |doi=10.1090/S0273-0979-1979-14551-8 |doi-access=free |mr=513750 |zbl=0414.20009 |s2cid=121953006}}

* {{Cite book |last1=Gorenstein |first1=D. |author1-link=Daniel Gorenstein |last2=Lyons |first2=Richard |last3=Solomon |first3=Ronald |title=The classification of the finite simple groups, Number 3 |url=https://bookstore.ams.org/surv-40-3/ |publisher=American Mathematical Society |location=Providence, R.I. |series=Mathematical Surveys and Monographs |pages=xiii, 1–362 |year=1998 |isbn=978-0-8218-0391-2 |oclc=6907721813 |doi=10.1112/S0024609398255457 |volume=40 |number=3 |mr=1490581 |s2cid=209854856}}

* {{Cite journal |author-link=Robert Griess |last=Griess, Jr. |first=Robert L. |title=The Friendly Giant |url=https://www.digizeitschriften.de/dms/img/?PPN=PPN356556735_0069&DMDID=dmdlog7 |journal=Inventiones Mathematicae |volume=69 |year=1982 |pages=1−102 |doi=10.1007/BF01389186 |bibcode=1982InMat..69....1G |hdl=2027.42/46608 |hdl-access=free |mr=671653 |s2cid=123597150 |zbl=0498.20013}}

* {{Cite book |last=Griess, Jr. |first=Robert L. |title=Twelve Sporadic Groups |series=Springer Monographs in Mathematics |publisher=Springer-Verlag |location=Berlin |year=1998 |pages=1−169 |isbn=9783540627784 |oclc=38910263 |mr=1707296 |zbl=0908.20007}}

* {{Citation |last1=Hartley |first1=Michael I. |last2=Hulpke |first2=Alexander |title=Polytopes Derived from Sporadic Simple Groups |url=https://cdm.ucalgary.ca/article/view/61945/46662 |journal=Contributions to Discrete Mathematics |volume=5 |number=2 |publisher=University of Calgary Department of Mathematics and Statistics |issn=1715-0868 |location=Alberta, CA |year=2010 |page=106−118 |doi=10.11575/cdm.v5i2.61945 |doi-access=free |mr=2791293 |zbl=1320.51021 |s2cid=40845205}}

* {{Cite journal |last=Hiss |first=Gerhard |title=Die Sporadischen Gruppen (The Sporadic Groups) |url=https://www.mathematik.de/images/DMV/Jahresberichte/Jahresberichte_Archiv/Jahresbericht_04-2003.pdf |journal=Jahresber. Deutsch. Math.-Verein. (Annual Report of the German Mathematicians Association) |volume=105 |issue=4 |year=2003 |pages=169−193 |issn=0012-0456 |mr=2033760 |zbl=1042.20007}} (German)

* {{Cite journal |last1=Howlett |first1=R. B. |last2=Rylands |first2=L. J. |last3=Taylor |first3=D. E. |title=Matrix generators for exceptional groups of Lie type |journal=Journal of Symbolic Computation |year=2001 |volume=31 |issue=4 |pages=429–445 |doi=10.1006/jsco.2000.0431 |doi-access=free |mr=1823074 |zbl=0987.20003 |s2cid=14682147}}

* {{Cite journal |last=Jansen |first=Christoph |date=2005 |title=The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups |journal=LMS Journal of Computation and Mathematics |volume=8 |pages=122−144 |publisher=London Mathematical Society |doi=10.1112/S1461157000000930 |doi-access=free |s2cid=121362819 |zbl=1089.20006 |mr=2153793}}

* {{Cite journal |last=Lubeck |first=Frank |title=Smallest degrees of representations of exceptional groups of Lie type |url=https://www.tandfonline.com/doi/abs/10.1081/AGB-100002175 |journal=Communications in Algebra |volume=29 |issue=5 |pages=2147−2169 |publisher=Taylor & Francis |location=Philadelphia, PA |year=2001 |doi=10.1081/AGB-100002175 |mr=1837968 |s2cid=122060727 |zbl=1004.20003|url-access=subscription }}

* {{Cite journal |last1=Nickerson |first1=S.J. |last2=Wilson |first2=R.A. |author2-link=Robert Anton Wilson |title=Semi-Presentations for the Sporadic Simple Groups |url=https://www.tandfonline.com/doi/abs/10.1080/10586458.2005.10128927 |journal=Experimental Mathematics |volume=14 |issue=3 |pages=359−371 |publisher=Taylor & Francis |year=2011 |location=Oxfordshire |doi=10.1080/10586458.2005.10128927 |mr=2172713 |zbl=1087.20025 |s2cid=13100616}}

* {{Cite book |last1=Ronan |first1=Mark |author-link=Mark Ronan |title=Symmetry and the Monster: One of the Greatest Quests of Mathematics |url=https://archive.org/details/symmetrymonstero0000rona |url-access=registration |publisher=Oxford University Press |location=New York |year=2006 |pages=vii, 1–255 |isbn=978-0-19-280722-9 |oclc=180766312 |mr=2215662 |zbl=1113.00002}}

* {{Cite web |editor-last=Sloane |editor-first=N. J. A. |editor-link=Neil Sloane |title=Orders of sporadic simple groups (A001228) |url=http://oeis.org/A001228 |website=The On-Line Encyclopedia of Integer Sequences |publisher=OEIS Foundation |year=1996}}

* {{Cite book |last=Wilson |first=R.A |author-link=Robert Arnott Wilson |title=The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249) |chapter=Chapter: An Atlas of Sporadic Group Representations |chapter-url=https://webspace.maths.qmul.ac.uk/r.a.wilson/pubs_files/ASGRweb.pdf |publisher=Cambridge University Press |location=Cambridge, U.K |year=1998 |pages=261–273 |doi=10.1017/CBO9780511565830.024 |isbn=9780511565830 |oclc=726827806 |mr=1647427 |zbl=0914.20016 |s2cid=59394831}}

* {{Cite web |last1=Wilson |first1=R.A. |author1-link=Robert Arnott Wilson |last2=Parker |first2=R.A. |author2-link=Richard A. Parker |last3=Nickerson |first3=S.J. |last4=Bray |first4=J.N. |title=ATLAS of Group Representations |website=ATLAS of Finite Group Representations |url=https://brauer.maths.qmul.ac.uk/Atlas/ |publisher=Queen Mary University of London |year=1999}}

{{Refend}}

== External links ==

* {{MathWorld|urlname=SporadicGroup|title=Sporadic Group}} * [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ Atlas of Finite Group Representations: Sporadic groups]

* Category:Mathematical tables

he:משפט המיון לחבורות פשוטות סופיות