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 that 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 (such as by John Conway) 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 M₁₁, M₁₂, M₂₂, M₂₃, M₂₄
• Janko groups J₁, J₂ or HJ, J₃ or HJM, J₄
• Conway groups Co₁ or F₂₋, 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 sporadic group Suz or F₃₋
• O'Nan group O'N
• Harada–Norton group HN or F₅₊ or F₅
• Lyons group Ly
• Tits group T
• Thompson group Th or F_3|3 or F₃
• Baby Monster group B or F₂₊ or F₂
• Fischer–Griess Monster group M or F₁

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".

Diagram is based on diagram given in 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 J₁, J₃, J₄, 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 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.

Second generation: the Leech lattice

The second generation are all 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: other subgroups of the Monster

The third generation 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 also belongs in this generation: there is a subgroup S₄ ײF₄(2)′ normalising a 2C² subgroup of B, giving rise to a subgroup 2·S₄ ײF₄(2)′ normalising a certain Q₈ subgroup of the Monster. ²F₄(2)′ is also a subgroup of the Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄′, and of the Baby Monster B. ²F₄(2)′ is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

Table of the sporadic group orders

References

• Burnside, William (1911), Theory of groups of finite order, pp. 504 (note N), ISBN 0486495752 (2004 reprinting) 
• 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 (volume 1), AMS, 1994 (volume 2), AMS.
• Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998.
• Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9 

External links

• Weisstein, Eric W., "Sporadic Group" from MathWorld.
• Atlas of Finite Group Representations: Sporadic groups