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 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, in which case the sporadic groups number 27.
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 who first predicted their existence. The full list is:
The Tits group T is sometimes also regarded as a sporadic group, which is why in some sources the number of sporadic groups is given as 27 instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type. Anyway, it is the of the infinite family of commutator groups all of them finite simple groups. For n>0 they coincide with the groups of Lie type But for the derived subgroup, called Tits group, has an index 2 in the group of Lie type.
Matrix representations over finite fields for all the sporadic groups have been constructed.
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".
The diagram at right is based on. 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.

I. Pariahs

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

II. 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

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:
Consists of subgroups which are closely related to the Monster group M:
The Tits group also belongs in this generation: there is a subgroup S4 ×2F4′ normalising a 2C2 subgroup of B, giving rise to a subgroup
2·S4 ×2F4′ normalising a certain Q8 subgroup of the Monster.
2F4′ is also a subgroup of the Fischer groups Fi22, Fi23 and Fi24′, and of the Baby Monster B.
2F4′ is also a subgroup of the Rudvalis group Ru, and has
no involvements in sporadic simple groups except the containments we have already mentioned.

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