Sporadic group
The sporadic groups are 26 special groups in group theory. These are all finite simple groups that do not fit into one of the 18 families of finite simple groups.
Discovery history
The first five discovered sporadic groups, the so-called Mathieugruppen were discovered by Émile Mathieu in the years 1862 and 1873. The discovery history of all other sporadic groups did not begin until 1964.
The earliest mention of the term "sporadic group " should of Burnside in 1911, referring to the Mathieugruppen already known at that time, come: These apparently sporadic simple groups would probably repay a closer examination than theyhave yet received.
Classification
20 of the 26 sporadic groups can be understood as subgroups or quotient groups of subgroups of the Monster group ( including the Monster group itself). These 20 groups are by Robert Griess as Happy Family ( German: Happy family ) designated. Except for the six groups are the groups Janko J1, J3 and J4 which O'Nan Group ( ON), the Rudvalisgruppe (Ru) and Lyon group ( Ly). These six exceptions are also called pariahs (English pariah ).
The Happy Family is divided naturally into three generations, the first generation with the extended binary Golay code and the second is connected to the Leech lattice and their automorphism groups. The first generation comprises the five Mathieugruppen the happy family, the second generation of the Conway groups Co1 to Co3, J2, McL, HS. The third generation includes the remaining sporadic groups.
Partly also named after the Belgian-French mathematician Jacques Tits Titsgruppe the order 17.9712 million is considered to be a sporadic group; Following this view, there would be 27 instead of 26 sporadic groups.