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Wikipedia / Sporadic group

In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The mentioned 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 26 exceptions are the sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups.

The monster group, or friendly giant, 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 Émile Mathieu in the 1860s and the other twenty-one were found between 1965 (J1) and 1975 (J4). 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:

Various constructions for these groups were first compiled in , including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at , updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in . A further exception in the classification of finite simple groups is the Tits group T, which is sometimes considered of Lie type or sporadic — 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.

The diagram at right is based on . It does not show the numerous non-sporadic simple subquotients of the sporadic groups. The earliest use of the term sporadic group may be where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.)

Organization

Happy Family

Of the 26 (or 27) sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups (sectionss). These twenty have been called the happy family by Robert Griess, and can be organized into three generations.

First generation (5 groups): the Mathieu groups

Mn 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 ) 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.)

Pariahs

The six sporadic groups which are not subquotients of the monster group M are J1, J3, J4, O'N, Ru, and Ly, sometimes known as the pariahs.

Tits group

If regarded as a sporadic group, the Tits group (T for short) would be a third generation group. There is a subgroup normalizing a subgroup of the Baby monster B, giving rise to a subgroup normalizing a certain Q8 subgroup of the Monster M. is also a subquotient of the Fischer group Fi22, and thus also of and , and of the Baby Monster B; involvement of T outside of third-generation sporadic groups, is only seen as being a subquotient of the (pariah) Rudvalis group Ru. T has no part in sporadic simple groups except the ones already mentioned. In terms of classification as a finite simple group, T is the of the infinite family of commutator groups , thus in a strict sense not sporadic, nor of Lie type (sources have historically regarded the Tits group as belonging in either grouping, or as neither sporadic nor of Lie type, or both altogether). For these finite simple groups coincide with the groups of Lie type , also known as Ree groups of type 2F4.

Assorted table of information

Here are listed the sporadic groups by their discovery date, and discoverer, alongside group generation (per Griess), order, minimal degree of faithful representation (dimensional Brauer character), and presentations via generators. The Tits group is included with a greyer shade of blue to reflect its ambiguous status within the classification scheme as a third-generation group.

Notes

References

Works cited

  • (German)

he:משפט המיון לחבורות פשוטות סופיות