Concept
Monster group
Related concepts (18)
Finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.
Lyons group
In the area of modern algebra known as group theory, the Lyons group Ly or Lyons-Sims group LyS is a sporadic simple group of order 283756711313767 = 51765179004000000 ≈ 5. Ly is one of the 26 sporadic groups and was discovered by Richard Lyons and Charles Sims in 1972-73. Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group A11 of degree 11 by the cyclic group C2.
Held group
In the area of modern algebra known as group theory, the Held group He is a sporadic simple group of order 21033527317 = 4030387200 ≈ 4. He is one of the 26 sporadic groups and was found by during an investigation of simple groups containing an involution whose centralizer is isomorphic to that of an involution in the Mathieu group M24. A second such group is the linear group L5(2). The Held group is the third possibility, and its construction was completed by John McKay and Graham Higman.
Rudvalis group
In the area of modern algebra known as group theory, the Rudvalis group Ru is a sporadic simple group of order 214335371329 = 145926144000 ≈ 1. Ru is one of the 26 sporadic groups and was found by and constructed by . Its Schur multiplier has order 2, and its outer automorphism group is trivial. In 1982 Robert Griess showed that Ru cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
Vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence. The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel.
Pariah group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group. The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family. For example, the orders of J4 and the Lyons Group Ly are divisible by 37. Since 37 does not divide the order of the monster, these cannot be subquotients of it; thus J4 and Ly are pariahs.
Thompson sporadic group
In the area of modern algebra known as group theory, the Thompson group Th is a sporadic simple group of order 2153105372131931 = 90745943887872000 ≈ 9. Th is one of the 26 sporadic groups and was found by and constructed by . They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E8. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E8(3).
Character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.