Pariah group
Encyclopedia
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the term pariah was introduced by to refer to the six sporadic simple groups that are not subquotients of the monster simple group.
These groups are:
- Three of the Janko groupJanko groupIn mathematics, a Janko group is one of the four sporadic simple groups named for Zvonimir Janko. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4...
s: J1Janko group J1In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...
, J3Janko group J3In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by , and it was shown to exist by...
, and J4Janko group J4In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko . Its existence and uniqueness was shown by Simon P. Norton and others in 1980...
. - The Lyons groupLyons groupIn the mathematical field of group theory, the Lyons group Ly , is a sporadic simple group of order...
(Ly) - The O'Nan groupO'Nan groupIn the mathematical field of group theory, the O'Nan group O'N is a sporadic simple group of orderThe Schur multiplier has order 3, and its outer automorphism group has order 2...
- The Rudvalis groupRudvalis groupIn the mathematical field of group theory, the Rudvalis group Ru is a sporadic simple group of order-Properties:The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group...