Sporadic group

The sporadic groups are 26 special groups in group theory . These are the finite simple groups that cannot be classified in one of the (18) families with an infinite number of members (of finite simple groups).

Discovery story

The first five sporadic groups discovered, the so-called Mathieu groups , were discovered by Émile Mathieu in 1862 and 1873. The history of discovery of all other sporadic groups did not begin until 1964.

The earliest mention of the term “sporadic group” is likely to come from Burnside 1911, referring to the Mathieu groups already known at the time: These apparently sporadic simple groups would probably repay a closer examination than they have yet received.

Classification

Hasse diagram of the 26 sporadic groups.
A line ascending from X to Y means:
X is a subquotient of Y
The circles indicate maximum subquotients; the 3 different squares belonging to the different generations.

In the adjacent Hasse diagram , two groups X and Y , which are related to each other in the relation X is the subquotient of Y, are connected by a line from X below to Y above. Groups without a line upwards, i.e. maximum in relation to the relation, are circled. Since the relation is transitive , implied connecting lines are omitted. Furthermore, for the sake of clarity, the diagram shows only simple groups X as sub-quotients , and only those for which there are no simple sub-quotients that really lie between X and Y.

20 of the 26 sporadic groups are subquotients of the monster group M, called by Robert Griess Friendly Giant (German: friendly giant ). According to Griess, these 20 groups are grouped under the name Happy Family (German: Happy Family ). The latter is divided into three generations, the first generation being related to the extended binary Golay code and the second to the Leech lattice or automorphism groups thereof. The five math groups belong to the first generation (parallelogram in the graphic), to the second generation the Conway groups (rhombus) Co ₁ to Co ₃ , J ₂ , McL, HS. The third generation (rectangle) is closely related to M and contains the other groups of the Happy Family .

The six sporadic groups that are not subquotients of the monster group are the Janko groups J ₁ , J ₃ and J ₄ , the O'Nan group (O'N), the Rudvalis group (Ru) and the Lyons group (Ly). They are at Griess pariah s (Engl. Pariah called) in the table (the generation -).

The Tits group T = ² F ₄ (2) ′ of order 17.971.200, named after the Belgian-French mathematician Jacques Tits , is also regarded as a sporadic group because it is not a group of the Lie type and therefore not Member of an infinite family of Lie-type groups. However, the definition for "non-sporadic" in finite simple groups is "belonging to an infinite family" - which has nothing to do with the property "of the Lie type". Because there are other infinite families of finite simple groups that are not of the Lie type either, for example the groups of prime order. With their membership of the infinite family ² F ₄ (2 ²ⁿ⁺¹ ) ′, whose members ² F ₄ (2 ²ⁿ⁺¹ ) ′ = ² F ₄ (2 ²ⁿ⁺¹ ) for coincide with their derivatives (and the are actually of the Lie type), they are not strictly a sporadic group. ${\ displaystyle n \ geq 1}$

Table of the 26 sporadic groups

Surname	Symbols	Explorer	year	generation	Order (approx)	Order (as decimal number sequence A001228 in OEIS )	Order (in prime factorization )
Math group M11	M ₁₁	Mathieu	1861	1	8the³	7,920	2 ⁴ 3 ² 5 11
Maths group M12	M ₁₂	Mathieu	1861	1	1e⁵	95.040	2 ⁶ 3 ³ 5 11
Math group M22	M ₂₂	Mathieu	1861	1	4the⁵	443,520	2 ⁷ 3 ² 5 7 11
Math group M23	M ₂₃	Mathieu	1861	1	1e^7th	10.200.960	2 ⁷ 3 ² 5 7 11 23
Math group M24	M ₂₄	Mathieu	1861	1	2e^8th	244.823.040	2 ¹⁰ 3 ³ 5 7 11 23
Jankogroup J1	J ₁	Janko	1964	-	2e⁵	175,560	2 ³ 3 5 7 11 19
Jankogroup J2	J ₂ , HJ	Janko	1966	2	6the⁵	604,800	2 ⁷ 3 ³ 5 ² 7
Jankogroup J3	J ₃	Janko	1966	-	5e^7th	50.232.960	2 ⁷ 3 ⁵ 5 17 19
Jankogroup J4	J ₄	Janko	1975	-	9e^19th	86,775,571,046,077,562,880	2 ²¹ 3 ³ 5 7 11 ³ 23 29 31 31 37 43
Higman Sims Group	HS	Higman , Sims	1967	2	4the^7th	44,352,000	2 ⁹ 3 ² 5 ³ 7 11
Conway group Co1	Co ₁ , C ₁	Conway	1968	2	4the^18th	4,157,776,806,543,360,000	2 ²¹ 3 ⁹ 5 ⁴ 7 ² 11 13 23
Conway group Co2	Co ₂ , C ₂	Conway	1969	2	4the¹³	42,305,421,312,000	2 ¹⁸ 3 ⁶ 5 ³ 7 11 23
Conway group Co3	Co ₃ , C ₃	Conway	1969	2	5e¹¹	495,766,656,000	2 ¹⁰ 3 ⁷ 5 ³ 7 11 23
Hero group	Hey	hero	1969	3	4the⁹	4,030,387,200	2 ¹⁰ 3 ³ 5 ² 7 ³ 17
McLaughlin Group	McL, Mc	McLaughlin	1969	2	9e^8th	898,128,000	2 ⁷ 3 ⁶ 5 ³ 7 11
Suzuki group	Suz	Suzuki	1969	2	4the¹¹	448.345.497.600	2 ¹³ 3 ⁷ 5 ² 7 11 13
Fischer Group FI22	Fi ₂₂ , M (22)	Fisherman	1976	3	6the¹³	64,561,751,654,400	2 ¹⁷ 3 ⁹ 5 ² 7 11 13
Fischer Group Fi23	Fi ₂₃ , M (23)	Fisherman	1976	3	4the^18th	4,089,470,473,293,004,800	2 ¹⁸ 3 ¹³ 5 ² 7 11 13 17 17 23
Fi24 fishing group	Fi ₂₄ , F ₂₄ ′, M (24)	Fisherman	1976	3	1e²⁴	1.255.205.709.190.661.721.292.800	2 ²¹ 3 ¹⁶ 5 ² 7 ³ 11 13 17 17 23 29
Lyons group	Ly	Lyons	1973	-	5e¹⁶	51,765,179,004,000,000	2 ⁸ 3 ⁷ 5 ⁶ 7 11 31 37 67
Rudvalis group	Ru	Rudvalis	1973	-	1e¹¹	145,926,144,000	2 ¹⁴ 3 ³ 5 ³ 7 13 29
Baby monsters group	B, F ₂	Fisherman	circa 1970	3	4the³³	4,154,781,481,226,426,191,177,580,544,000,000	2 ⁴¹ · 3 ¹³ · 5 ⁶ · 7 ² · 11 · 13 · 17 · 19 · 23 · 31 · 47
O'Nan group	O'N	O'Nan	1976	-	4the¹¹	460.815.505.920	2 ⁹ 3 ⁴ 5 7 ³ 11 19 31
Thompson group	Th, F ₃	Thompson	1976	3	9e¹⁶	90,745,943,887,872,000	2 ¹⁵ 3 ¹⁰ 5 ³ 7 ² 13 19 31
Harada Norton Group	HN, F ₅	Harada , Norton , Smith	1976	3	3e^14th	273,030,912,000,000	2 ¹⁴ 3 ⁶ 5 ⁶ 7 11 19
Monster group	M, F ₁	Fischer , Griess	1976	3	8the⁵³	808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000	2 ⁴⁶ 3 ²⁰ 5 ⁹ 7 ⁶ 11 ² 13 ³ 17 19 23 29 31 41 47 47 59 71

literature

Robert Griess : Twelve Sporadic Groups . Springer, 2002, ISBN 978-3-540-62778-4 , doi : 10.1007 / 978-3-662-03516-0 .
Robert Griess: The Friendly Giant . In: Inventiones Mathematicae . tape 69 , 1982, pp. 1–102 , doi : 10.1007 / BF01389186 ( online at digizeitschriften.de ).
John McKay : Finite Groups - Coming of Age . American Mathematical Society, 1985, ISBN 978-0-8218-5047-3 .
Michael Aschbacher : Sporadic Groups , Cambridge University Press 1994

Individual evidence

↑ compiled mainly from Griess2 p. 94
↑ F ₁ in Griess2
↑ s. Semolina 2
↑ In Eric W. Weisstein "Sporadic Group" From MathWorld - A Wolfram Web Resource , the Tits group is not listed under the 26. However, there is still a link to the sporadic groups in Eric W. Weisstein's "Tits Group" From MathWorld - A Wolfram Web Resource .
↑ The first symbol is the only one listed in the #Atlas .

Web links

The sporadic groups (producers, subgroups, conjugacy ...) in Atlas of Finite Group Representations (English)

[1] y from Griess2 p. 94

[2] F ₁ in Griess2

[3] s. Semolina 2

[4] In Eric W. Weisstein "Sporadic Group" From MathWorld - A Wolfram Web Resource , the Tits group is not listed under the 26. However, there is still a link to the sporadic groups in Eric W. Weisstein's "Tits Group" From MathWorld - A Wolfram Web Resource .

[5] The first symbol is the only one listed in the #Atlas .