Sporadic group

The sporadic group is one of 26 exceptional groups in the classification theorem for simple finite groups .

A simple group is a group G that does not contain any normal subgroups other than the group G itself and a trivial (unit) subgroup. The classification theorem states that the consists of 18 countable infinite families, plus 26 exceptions that do not fall into this classification. These exceptions are called sporadic groups. They are also known as “sporadic simple groups” or “sporadic end groups”. Since the Tits group is not strictly a group of Lie type , sometimes it is also considered sporadic ^[1] and in this case is the 27th sporadic group.

The Monster group is the largest among sporadic groups and contains all but six of the other sporadic groups as subgroups or groups.

Content

Names of sporadic groups

Five sporadic groups were discovered by Mathieu in the 1860s, the remaining 21 were found between 1965 and 1975. The existence of several of these groups was predicted before they were built. It was later proved that this completely completed the full search. Most groups bear the names of the mathematicians who first predicted their existence. Full list of groups:

The diagram shows the sub-factor relationships of sporadic groups.

Mathieu groups , , , ,
Janko groups , J ₂ or HJ , ,
Conway groups Co ₁ , ,
Fisher groups , ,
HS
McL
He or F ₇₊ , or F ₇
Ru
Suz or F ₃₋
O'N
HN or F ₅₊ or F ₅
Ly
Th or F _{3 | 3} , or F ₃
B or F ₂₊ , or F ₂
Fisher-Grace Monster Group M or F ₁

The Tits group T is sometimes also considered a sporadic group (it is almost a Lie type) and for this reason, according to some sources, the number of sporadic groups is given as 27 rather than 26. According to other sources, the Tits group is not considered either a sporadic or a group of Lie type.

For all sporadic groups, matrix representations over finite fields were constructed.

The earliest use of the term “sporadic group” was found in Burnside ^[2] , where he speaks of Mathieu groups: “These apparently sporadic simple groups require more thorough investigation than they have received so far.”

The diagram on the right is based on the Ronan diagram ^[3] . Sporadic groups also have a large number of subgroups that are not sporadic, but they are not represented on the diagram because of their huge number.

System

Of the 26 sporadic groups, 20 are inside the Monster group as subgroups or .

I. Pariah

The six exceptions J ₁ , J ₃ , J ₄ , O'N , Ru, and Ly are sometimes called .

II. Happy Family

The remaining twenty groups are called the Happy Family (the name was given by ) and can be divided into three generations.

First generation (5 groups) - Mathieu groups

The groups M _n for n = 11, 12, 22, 23, and 24 are multiple-transitive groups of permutations of n points. All of them are subgroups of the group M ₂₄ , which is a group of permutations of 24 points.

Second Generation (7 Groups) - Lich Lattice

All group of automorphisms of a lattice in a 24-dimensional space called the Lich lattice :

Co ₁ is the factor group of the automorphism group in the center {± 1}
Co ₂ is a stabilizer of type 2 vector (i.e., length 2)
Co ₃ is a stabilizer of type 3 vector (i.e., length √6)
Suz - a group of automorphisms that preserve the structure (center module)
McL - Type 2-2-3 Triangle Stabilizer
HS - type 2-3-3 triangle stabilizer
J ₂ is a group of automorphisms preserving a quaternionic structure (module in the center).

Third Generation (8 Groups) - Other Monster Subgroups

Consists of subgroups that are closely related to Monster M :

B or F ₂ has a double coating, which is the centralizer of an element of order 2 in M
Fi ₂₄ ′ has a triple cover, which is a centralizer of an element of order 3 in M ( conjugacy class “3A”)
Fi ₂₃ is a subgroup of Fi ₂₄ ′
Fi ₂₂ has a double coating, which is a subgroup of Fi ₂₃
The product Th = F ₃ and a group of order 3 is the centralizer of an element of order 3 in M ( 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 itself is considered to belong to 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 also belongs to this generation - there is a subgroup $S_{4}\times {^{2}}{F_{4}(2)^{\prime }}$ ${\ displaystyle S_ {4} \ times {^ {2}} {F_ {4} (2) ^ {\ prime}}}$ normalizing the 2C ² subgroup B generating the subgroup $2{\cdot }S_{4}\times {^{2}}F_{4}(2)^{\prime }$ ${\ displaystyle 2 {\ cdot} S_ {4} \ times {^ {2}} F_ {4} (2) ^ {\ prime}}$ normalizing some subgroup Q _{8 of the} Monster. ${^{2}}F_{4}(2)^{\prime }$ ${\ displaystyle {^ {2}} F_ {4} (2) ^ {\ prime}}$ is also a subgroup of the Fisher groups Fi ₂₂ , Fi ₂₃ and Fi ₂₄ ′ and the “small Monster” B. ${^{2}}F_{4}(2)^{\prime }$ ${\ displaystyle {^ {2}} F_ {4} (2) ^ {\ prime}}$ It is a subgroup of the Rudvalis Ru pariah group and has no other dependencies with sporadic simple groups other than those listed above.

Sporadic group order table

Group	Generation	Order A001228 sequence in OEIS	Meaningful numbers	Decomposition	Troika Standard generators (a, b, ab) ^[4] ^[5] ^[6]	Other conditions
F ₁ or M	the third	8080174247945128758864599049617107 57005754368000000000	≈ 8⋅10 ⁵³	2 ⁴⁶ • 3 ²⁰ • 5 ⁹ • 7 ⁶ • 11 ² • 13 ³ • 17 • 19 • 23 • 29 • 31 • 41 • 47 • 59 • 71	2A, 3B, 29
	the third	4154781481226426191177580544000000	≈ 4⋅10 ³³	$2^{41}\cdot 3^{13}\cdot 5^{6}\cdot 7^{2}\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47$ {\ displaystyle 2 ^ {41} \ cdot 3 ^ {13} \ cdot 5 ^ {6} \ cdot 7 ^ {2} \ cdot 11 \ cdot 13 \ cdot 17 \ cdot 19 \ cdot 23 \ cdot 31 \ cdot 47 }	2C, 3A, 55	$o((ab)^{2}(abab^{2})^{2}ab^{2})=23$ ${\ displaystyle o ((ab) ^ {2} (abab ^ {2}) ^ {2} ab ^ {2}) = 23}$
	the third	1255205709190661721292800	≈ 1⋅10 ²⁴	2 ²¹ • 3 ¹⁶ • 5 ² • 7 ³ • 11 • 13 • 17 • 23 • 29	2A, 3E, 29	$o((ab)^{3}b)=33$ ${\ displaystyle o ((ab) ^ {3} b) = 33}$
	the third	4089470473293004800	≈ 4⋅10 ¹⁸	2 ¹⁸ • 3 ¹³ • 5 ² • 7 • 11 • 13 • 17 • 23	2B, 3D, 28
	the third	64561751654400	≈ 6⋅10 ¹³	2 ¹⁷ • 3 ⁹ • 5 ² • 7 • 11 • 13	2A, 13, 11	$o((ab)^{2}(abab^{2})^{2}ab^{2})=12$ ${\ displaystyle o ((ab) ^ {2} (abab ^ {2}) ^ {2} ab ^ {2}) = 12}$
	the third	90745943887872000	≈ 9⋅10 ¹⁶	2 ¹⁵ • 3 ¹⁰ • 5 ³ • 7 ² • 13 • 19 • 31	2, 3A, 19
	pariah	51765179004000000	≈ 5⋅10 ¹⁶	2 ⁸ • 3 ⁷ • 5 ⁶ • 7 • 11 • 31 • 37 • 67	2, 5A, 14	$o(ababab^{2})=67$ ${\ displaystyle o (ababab ^ {2}) = 67}$
	the third	273030912000000	≈ 3⋅10 ¹⁴	2 ¹⁴ • 3 ⁶ • 5 ⁶ • 7 • 11 • 19	2A, 3B, 22	$o([a,b])=5$ ${\ displaystyle o ([a, b]) = 5}$
Co ₁	second	4157776806543360000	≈ 4⋅10 ¹⁸	2 ²¹ • 3 ⁹ • 5 ⁴ • 7 ² • 11 • 13 • 23	2B, 3C, 40
	second	42305421312000	≈ 4⋅10 ¹³	2 ¹⁸ • 3 ⁶ • 5 ³ • 7 • 11 • 23	2A, 5A, 28
	second	495766656000	≈ 5⋅10 ¹¹	2 ¹⁰ • 3 ⁷ • 5 ³ • 7 • 11 • 23	2A, 7C, 17
	pariah	460815505920	≈ 5⋅10 ¹¹	2 ⁹ • 3 ⁴ • 5 • 7 ³ • 11 • 19 • 31	2A, 4A, 11
	second	448345497600	≈ 4⋅10 ¹¹	2 ¹³ • 3 ⁷ • 5 ² • 7 • 11 • 13	2B, 3B, 13	$o([a,b])=15$ ${\ displaystyle o ([a, b]) = 15}$
	pariah	145926144000	≈ 1⋅10 ¹¹	2 ¹⁴ • 3 ³ • 5 ³ • 7 • 13 • 29	2B, 4A, 13
	the third	4030387200	≈ 4⋅10 ⁹	2 ¹⁰ • 3 ³ • 5 ² • 7 ³ • 17	2A, 7C, 17
	second	898128000	≈ 9⋅10 ⁸	2 ⁷ • 3 ⁶ • 5 ³ • 7 • 11	2A, 5A, 11	$o((ab)^{2}(abab^{2})^{2}ab^{2})=7$ ${\ displaystyle o ((ab) ^ {2} (abab ^ {2}) ^ {2} ab ^ {2}) = 7}$
	second	44352000	≈ 4⋅10 ⁷	2 ⁹ • 3 ² • 5 ³ • 7 • 11	2A, 5A, 11
	pariah	86775571046077562880	≈ 9⋅10 ¹⁹	2 ²¹ • 3 ³ • 5 • 7 • 11 ³ • 23 • 29 • 31 • 37 • 43	2A, 4A, 37	$o(abab^{2})=10$ ${\ displaystyle o (abab ^ {2}) = 10}$
	pariah	50232960	≈ 5⋅10 ⁷	2 ⁷ • 3 ⁵ • 5 • 17 • 19	2A, 3A, 19	$o([a,b])=9$ ${\ displaystyle o ([a, b]) = 9}$
J ₂ or HJ	second	604800	≈ 6⋅10 ⁵	2 ⁷ • 3 ³ • 5 ² • 7	2B, 3B, 7	$o([a,b])=12$ ${\ displaystyle o ([a, b]) = 12}$
	pariah	175560	≈ 2⋅10 ⁵	2 ³ • 3 • 5 • 7 • 11 • 19	2, 3, 7	$o(abab^{2})=19$ ${\ displaystyle o (abab ^ {2}) = 19}$
	the first	244823040	≈ 2⋅10 ⁸	2 ¹⁰ • 3 ³ • 5 • 7 • 11 • 23	2B, 3A, 23	$o(ab(abab^{2})^{2}ab^{2})=4$ ${\ displaystyle o (ab (abab ^ {2}) ^ {2} ab ^ {2}) = 4}$
	the first	10200960	≈ 1⋅10 ⁷	2 ⁷ • 3 ² • 5 • 7 • 11 • 23	2, 4, 23	$o((ab)^{2}(abab^{2})^{2}ab^{2})=8$ ${\ displaystyle o ((ab) ^ {2} (abab ^ {2}) ^ {2} ab ^ {2}) = 8}$
	the first	443520	≈ 4⋅10 ⁵	2 ⁷ • 3 ² • 5 • 7 • 11	2A, 4A, 11	$o(abab^{2})=11$ ${\ displaystyle o (abab ^ {2}) = 11}$
	the first	95040	≈ 1⋅10 ⁵	2 ⁶ • 3 ³ • 5 • 11	2B, 3B, 11
	the first	7920	≈ 8⋅10 ³	2 ⁴ • 3 ² • 5 • 11	2, 4, 11	$o((ab)^{2}(abab^{2})^{2}ab^{2})=4$ ${\ displaystyle o ((ab) ^ {2} (abab ^ {2}) ^ {2} ab ^ {2}) = 4}$

Notes

↑ For example, according to Conway .
↑ Burnside, 1911 , p. 504, note N.
↑ Ronan, 2006 .
↑ An Atlas of Sporadic Group Representations (Neopr.) (1998).
↑ Semi-Presentations for the Sporadic Simple Groups (Neopr.) (2000).
↑ Atlas: Sporadic Groups (Neopr.) (1999).

Literature

William Burnside. Theory of groups of finite order. - 1911. - S. 504 (note N). - ISBN 0-486-49575-2 .
Conway JH A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups // Proc. Natl. Acad. Sci. USA - 1968. - T. 61 , no. 2 . - S. 398-400 . - DOI : 10.1073 / pnas . 61.2.398 .
Conway JH , Curtis RT, Norton SP, Wilson RA Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from JG Thackray. - Oxford University Press, 1985. - ISBN 0-19-853199-0 .
Gorenstein D., Lyons R., Solomon R. The Classification of the Finite Simple Groups. - American Mathematical Society , 1994. Issues 1 , 2 , ...
Robert L. Griess. Twelve Sporadic Groups . - Springer-Verlag, 1998 .-- ISBN 3540627782 .
Mark Ronan Symmetry and the Monster . - Oxford, 2006. - ISBN 978-0-19-280722-9 .

Links

Weisstein, Eric W. Sporadic Group on Wolfram MathWorld .
Atlas of Finite Group Representations: Sporadic groups

[1] For example, according to Conway .

[_88f68378ebe399ea-2] Burnside, 1911 , p. 504, note N.

[_a05897571a34b30f-3] Ronan, 2006 .

[4] An Atlas of Sporadic Group Representations (Neopr.) (1998).

[5] Semi-Presentations for the Sporadic Simple Groups (Neopr.) (2000).

[6] Atlas: Sporadic Groups (Neopr.) (1999).