Sporadic group

In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. 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. 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. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type. Anyway, it is the (n=0)-member 2F4(2)′ of the infinite family of commutator groups 2F4(22n+1)′, all of them finite simple groups. For n>0 they coincide with the groups of Lie type 2F4(22n+1). But for n=0, the derived subgroup 2F4(2)′, called Tits group, has an index 2 in the group 2F4(2) 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 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).