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
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 2 n +1 ) ′, whose members 2 F 4 (2 2 n +1 ) ′ = 2 F 4 (2 2 n +1 ) for coincide with their derivatives (and the are actually of the Lie type), they are not strictly a sporadic group.
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 | 1e7th | 10.200.960 | 2 7 3 2 5 7 11 23 |
Math group M24 | M 24 | Mathieu | 1861 | 1 | 2e8th | 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 | - | 5e7th | 50.232.960 | 2 7 3 5 5 17 19 |
Jankogroup J4 | J 4 | Janko | 1975 | - | 9e19th | 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 | 4the7th | 44,352,000 | 2 9 3 2 5 3 7 11 |
Conway group Co1 | Co 1 , C 1 | Conway | 1968 | 2 | 4the18th | 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 | 9e8th | 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 | 4the18th | 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 | 3e14th | 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)