Sporadic group

In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families[1] plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group,[2] in which case there would be 27 sporadic groups.

The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.

Names

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:

The diagram shows the subquotient relations between the sporadic groups. A connecting line means the lower group is a subquotient of the upper, with no sporadic subquotient in between.

1st generation,

2nd generation,

3rd generation,

Pariah

Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
Conway groups Co₁, Co₂, Co₃
Fischer groups Fi₂₂, Fi₂₃, Fi₂₄′ or F₃₊
Higman–Sims group HS
McLaughlin group McL
Held group He or F₇₊ or F₇
Rudvalis group Ru
Suzuki group Suz or F₃₋
O'Nan group O'N
Harada–Norton group HN or F₅₊ or F₅
Lyons group Ly
Thompson group Th or F_3|3 or F₃
Baby Monster group B or F₂₊ or F₂
Fischer–Griess Monster group M or F₁

The Tits group T is sometimes also regarded as a sporadic group (it is almost but not strictly a group of Lie type), which is why in some sources the number of sporadic groups is given as 27 instead of 26.[3] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.[4] Anyway, it is the (n = 0)-member ²F₄(2)′ of the infinite family of commutator groups ²F₄(2²ⁿ⁺¹)′ — and thus per definitionem not sporadic. For n > 0 these finite simple groups coincide with the groups of Lie type ²F₄(2²ⁿ⁺¹). But for n = 0, the derived subgroup ²F₄(2)′, called Tits group, is simple and has an index 2 in the finite group ²F₄(2) of Lie type which —as the only one of the whole family— is not simple.

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

The earliest use of the term sporadic group may be Burnside (1911, 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."

The diagram at right is based on Ronan (2006). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups (sections).

Happy family

The remaining twenty have been called the happy family by Robert Griess, and can be organized into three generations.

First generation (5 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.

Second generation (7 groups): the Leech lattice

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:

Co₁ is the quotient of the automorphism group by its center {±1}
Co₂ is the stabilizer of a type 2 (i.e., length 2) vector
Co₃ 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
J₂ 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:

B or F₂ has a double cover which is the centralizer of an element of order 2 in M
Fi₂₄′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
Fi₂₃ is a subgroup of Fi₂₄′
Fi₂₂ has a double cover which is a subgroup of Fi₂₃
The product of Th = F₃ 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 = F₅ and a group of order 5 is the centralizer of an element of order 5 in M
The product of He = F₇ 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 M₁₂ 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 ×²F₄(2)′ normalising a 2C² subgroup of B, giving rise to a subgroup 2·S₄ ×²F₄(2)′ normalising a certain Q₈ subgroup of the Monster. ²F₄(2)′ is also a subquotient of the Fischer group Fi₂₂, and thus also of Fi₂₃ and Fi₂₄′, 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

The six exceptions are J₁, J₃, J₄, O'N, Ru and Ly, sometimes known as the pariahs.

Table of the sporadic group orders (w/ Tits group)

Group	Gen.	Order, OEIS A001228		Factorized order	Standard generators triple (a, b, ab)[5][6][3]	Further conditions
F₁ or M	3rd	808017424794512875886459904961710757005754368000000000	≈ 8×10⁵³	2⁴⁶ · 3²⁰ · 5⁹ · 7⁶ · 11² · 13³ · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71	2A, 3B, 29	None
F₂ or B	3rd	4154781481226426191177580544000000	≈ 4×10³³	2⁴¹ · 3¹³ · 5⁶ · 7² · 11 · 13 · 17 · 19 · 23 · 31 · 47	2C, 3A, 55	$o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=23$
Fi₂₄' or F₃₊	3rd	1255205709190661721292800	≈ 1×10²⁴	2²¹ · 3¹⁶ · 5² · 7³ · 11 · 13 · 17 · 23 · 29	2A, 3E, 29	$o\left((ab)^{3}b\right)=33$
Fi₂₃	3rd	4089470473293004800	≈ 4×10¹⁸	2¹⁸ · 3¹³ · 5² · 7 · 11 · 13 · 17 · 23	2B, 3D, 28	None
Fi₂₂	3rd	64561751654400	≈ 6×10¹³	2¹⁷ · 3⁹ · 5² · 7 · 11 · 13	2A, 13, 11	$o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=12$
F₃ or Th	3rd	90745943887872000	≈ 9×10¹⁶	2¹⁵ · 3¹⁰ · 5³ · 7² · 13 · 19 · 31	2, 3A, 19	None
Ly	Pariah	51765179004000000	≈ 5×10¹⁶	2⁸ · 3⁷ · 5⁶ · 7 · 11 · 31 · 37 · 67	2, 5A, 14	$o\left(ababab^{2}\right)=67$
F₅ or HN	3rd	273030912000000	≈ 3×10¹⁴	2¹⁴ · 3⁶ · 5⁶ · 7 · 11 · 19	2A, 3B, 22	$o([a,b])=5$
Co₁	2nd	4157776806543360000	≈ 4×10¹⁸	2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23	2B, 3C, 40	None
Co₂	2nd	42305421312000	≈ 4×10¹³	2¹⁸ · 3⁶ · 5³ · 7 · 11 · 23	2A, 5A, 28	None
Co₃	2nd	495766656000	≈ 5×10¹¹	2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23	2A, 7C, 17	None
O'N	Pariah	460815505920	≈ 5×10¹¹	2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31	2A, 4A, 11	None
Suz	2nd	448345497600	≈ 4×10¹¹	2¹³ · 3⁷ · 5² · 7 · 11 · 13	2B, 3B, 13	$o([a,b])=15$
Ru	Pariah	145926144000	≈ 1×10¹¹	2¹⁴ · 3³ · 5³ · 7 · 13 · 29	2B, 4A, 13	None
F₇ or He	3rd	4030387200	≈ 4×10⁹	2¹⁰ · 3³ · 5² · 7³ · 17	2A, 7C, 17	None
McL	2nd	898128000	≈ 9×10⁸	2⁷ · 3⁶ · 5³ · 7 · 11	2A, 5A, 11	$o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=7$
HS	2nd	44352000	≈ 4×10⁷	2⁹ · 3² · 5³ · 7 · 11	2A, 5A, 11	None
J₄	Pariah	86775571046077562880	≈ 9×10¹⁹	2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43	2A, 4A, 37	$o\left(abab^{2}\right)=10$
J₃ or HJM	Pariah	50232960	≈ 5×10⁷	2⁷ · 3⁵ · 5 · 17 · 19	2A, 3A, 19	$o([a,b])=9$
J₂ or HJ	2nd	604800	≈ 6×10⁵	2⁷ · 3³ · 5² · 7	2B, 3B, 7	$o([a,b])=12$
J₁	Pariah	175560	≈ 2×10⁵	2³ · 3 · 5 · 7 · 11 · 19	2, 3, 7	$o\left(abab^{2}\right)=19$
T	3rd	17971200	≈ 2×10⁷	2¹¹ · 3³ · 5² · 13	2A, 3, 13	$o([a,b])=5$
M₂₄	1st	244823040	≈ 2×10⁸	2¹⁰ · 3³ · 5 · 7 · 11 · 23	2B, 3A, 23	$o\left(ab\left(abab^{2}\right)^{2}ab^{2}\right)=4$
M₂₃	1st	10200960	≈ 1×10⁷	2⁷ · 3² · 5 · 7 · 11 · 23	2, 4, 23	$o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=8$
M₂₂	1st	443520	≈ 4×10⁵	2⁷ · 3² · 5 · 7 · 11	2A, 4A, 11	$o\left(abab^{2}\right)=11$
M₁₂	1st	95040	≈ 1×10⁵	2⁶ · 3³ · 5 · 11	2B, 3B, 11	None
M₁₁	1st	7920	≈ 8×10³	2⁴ · 3² · 5 · 11	2, 4, 11	$o\left((ab)^{2}\left(abab^{2}\right)^{2}ab^{2}\right)=4$

References

The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups ²F₄(2²ⁿ⁺¹)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
For example, by John Conway.
Wilson RA, Parker RA, Nickerson SJ, Bray JN (1999). "Atlas: Sporadic Groups".
In Eric W. Weisstein „Tits Group“ From MathWorld--A Wolfram Web Resource there is a link from the Tits group to „Sporadic Group“, whereas in Eric W. Weisstein „Sporadic Group“ From MathWorld--A Wolfram Web Resource, however, the Tits group is not listed among the 26. Both sources checked on 2018-05-26.
Wilson RA (1998). "An Atlas of Sporadic Group Representations" (PDF).
Nickerson SJ, Wilson RA (2000). "Semi-Presentations for the Sporadic Simple Groups".

Burnside, William (1911), Theory of groups of finite order, p. 504 (note N), ISBN 0-486-49575-2
Conway, J. H. (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proc. Natl. Acad. Sci. U.S.A., 61 (2): 398–400, doi:10.1073/pnas.61.2.398, PMC 225171, PMID 16591697, Zbl 0186.32401
Griess, Robert L. (1982), "The Friendly Giant", Inventiones Mathematicae, 69, p. 1−102, doi:10.1007/BF01389186, hdl:2027.42/46608
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press. ISBN 0-19-853199-0. Zbl 0568.20001.
Gorenstein, D.; Lyons, R.; Solomon, R. (1994), The Classification of the Finite Simple Groups, American Mathematical Society Issues 1, 2, ...
Griess, Robert L. (1998), Twelve Sporadic Groups, Springer-Verlag, ISBN 3540627782, Zbl 0908.20007
Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9, Zbl 1113.00002

External links

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[1] The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups ²F₄(2²ⁿ⁺¹)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.

[2] For example, by John Conway.

[brauer-3] Wilson RA, Parker RA, Nickerson SJ, Bray JN (1999). "Atlas: Sporadic Groups".

[4] In Eric W. Weisstein „Tits Group“ From MathWorld--A Wolfram Web Resource there is a link from the Tits group to „Sporadic Group“, whereas in Eric W. Weisstein „Sporadic Group“ From MathWorld--A Wolfram Web Resource, however, the Tits group is not listed among the 26. Both sources checked on 2018-05-26.

[5] Wilson RA (1998). "An Atlas of Sporadic Group Representations" (PDF).

[6] Nickerson SJ, Wilson RA (2000). "Semi-Presentations for the Sporadic Simple Groups".