Sporadic group

Sporadic group

In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group "G" that does not have any normal subgroups except for the subgroup consisting only of the identity element, and "G" itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions which do not follow such a systematic pattern. These are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Sometimes the Tits group is regarded as a sporadic group (because it is not strictly a group of Lie type), in which case there are 27 sporadic groups.

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

Names of the sporadic groups

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. 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:
* 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 "Co"1 or "F"2−, "Co"2, "Co"3
* Fischer groups "Fi"22, "Fi"23, "Fi"24′ or "F"3+
* Higman-Sims group "HS"
* McLaughlin group "McL"
* Held group "He" or "F"7+ or "F"7
* Rudvalis group "Ru"
* Suzuki sporadic group "Suz" or "F"3−
* O'Nan group "O'N"
* 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

Matrix representations over finite fields for all the sporadic groups have been computed.

The earliest use of the term "sporadic group" may be harvtxt|Burnside|1911|loc=p. 504, note N where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received".

Diagram is based on diagram given in Harvtxt|Ronan|2006. The sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous.

Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups. The six exceptions are "J"1, "J"3, "J"4, "O'N", "Ru" and "Ly". These six groups are sometimes known as the pariahs.

The remaining twenty groups have been called the "Happy Family" by Robert Griess, and can be organized in into three generations.

First generation: the Mathieu groups

The 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.

econd generation: the Leech lattice

The second generation are all subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:
* "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 √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: other subgroups of the Monster

The third generation consists of subgroups which are closely related to the Monster group "M":
* "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 also belongs in this generation: there is a subgroup S4 ×2F4(2)′ normalising a 2C2 subgroup of "B", giving rise to a subgroup2·S4 ×2F4(2)′ normalising a certain Q8 subgroup of the Monster.2F4(2)′ is also a subgroup of the Fischer groups "Fi"22, "Fi"23 and "Fi"24′, and of the Baby Monster "B".

Table of the sporadic group orders

GroupOrder OEIS|id=A0012281SFFactorized order
"F"1 or "M" 808017424794512875886459904961710757005754368000000000 ≈ 8e|53 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
"F"2 or "B" 4154781481226426191177580544000000 ≈ 4e|33 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
"Fi"24' or "F"3+ 1255205709190661721292800 ≈ 1e|24 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29
"Fi"23 4089470473293004800 ≈ 4e|18 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23
"Fi"22 64561751654400 ≈ 6e|13 217 · 39 · 52 · 7 · 11 · 13
"F"3 or "Th" 90745943887872000 ≈ 9e|16215 · 310 · 53 · 72 · 13 · 19 · 31
"Ly" 51765179004000000 ≈ 5e|16 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
"F"5 or "HN" 273030912000000 ≈ 3e|14 214 · 36 · 56 · 7 · 11 · 19
"Co"1 4157776806543360000 ≈ 4e|18 221 · 39 · 54 · 72 · 11 · 13 · 23
"Co"2 42305421312000 ≈ 4e|13 218 · 36 · 53 · 7 · 11 · 23
"Co"3 495766656000 ≈ 5e|11 210 · 37 · 53 · 7 · 11 · 23
"O'N" 460815505920 ≈ 5e|11 29 · 34 · 5 · 73 · 11 · 19 · 31
"Suz" 448345497600 ≈ 4e|11 213 · 37 · 52 · 7 · 11 · 13
"Ru" 145926144000 ≈ 1e|11 214 · 33 · 53 · 7 · 13 · 29
"He" 4030387200 ≈ 4e|9 210 · 33 · 52 · 73 · 17
"McL" 898128000 ≈ 9e|8 27 · 36 · 53 · 7 · 11
"HS" 44352000 ≈ 4e|7 29 · 32 · 53 · 7 · 11
"J"4 86775571046077562880 ≈ 9e|19 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43
"J"3 or "HJM" 50232960 ≈ 5e|7 27 · 35 · 5 · 17 · 19
"J"2 or "HJ" 604800 ≈ 6e|5 27 · 33 · 52 · 7
"J"1 175560 ≈ 2e|5 23 · 3 · 5 · 7 · 11 · 19
"M"24 244823040 ≈ 2e|8 210 · 33 · 5 · 7 · 11 · 23
"M"23 10200960 ≈ 1e|727 · 32 · 5 · 7 · 11 · 23
"M"22443520 ≈ 4e|5 27 · 32 · 5 · 7 · 11
"M"1295040 ≈ 1e|526 · 33 · 5 · 11
"M"117920 ≈ 8e|324 · 32 · 5 · 11

References

*citation|first=William|last=Burnside|isbn=0486495752 (2004 reprinting)|year=1911|title=Theory of groups of finite order|pages=504 (note N)
* Conway, J. H.: "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398-400.
* Conway, J. H.: Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., "Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray." Eynsham: Oxford University Press, 1985, ISBN 0-19-853199-0
* Daniel Gorenstein, Richard Lyons, Ronald Solomon "The Classification of the Finite Simple Groups" [http://www.ams.org/online_bks/surv401/ (volume 1)] , AMS, 1994 [http://www.ams.org/online_bks/surv402/ (volume 2)] , AMS.
* Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.
*

External links

*
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ Atlas of Finite Group Representations: Sporadic groups]


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