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.

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 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.

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

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.

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

Consists of subgroups which are closely related to the Monster group M:

(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.

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