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In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M 11, M 12, M 22, M 23 and M 24 introduced by Mathieu (1861, 1873). They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects.
In the area of modern algebra known as group theory, the Mathieu group M 11 is a sporadic simple group of order 7,920 = 11 · 10 · 9 · 8 = 2 4 · 3 2 · 5 · 11. History and properties
The theory of umbral moonshine is a partly conjectural relationship between K3 surfaces and M 24. The Conway group Co1, the Fischer group Fi24, and the Janko group J4 each have maximal subgroups that are an extension of the Mathieu group M 24 by a group 2 11. (These extensions are not all the same.) [citation needed]
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
Earlier, Alfred Tarski proved elementary group theory undecidable. [31] The period of 1960-1980 was one of excitement in many areas of group theory. In finite groups, there were many independent milestones. One had the discovery of 22 new sporadic groups, and the completion of the first generation of the classification of finite simple groups.
The prehistory of Mathieu moonshine starts with a theorem of Mukai, asserting that any group of symplectic automorphisms of a K3 surface embeds in the Mathieu group M23. The moonshine observation arose from physical considerations: any K3 sigma-model conformal field theory has an action of the N=(4,4) superconformal algebra , arising from a ...
This is the group obtained from the orthogonal group in dimension 2n + 1 by taking the kernel of the determinant and spinor norm maps. B 1 (q) also exists, but is the same as A 1 (q). B 2 (q) has a non-trivial graph automorphism when q is a power of 2. This group is obtained from the symplectic group in 2n dimensions by quotienting out the center.
The automorphism group of the extended binary Golay code is the Mathieu group, of order 2 10 × 3 3 × 5 × 7 × 11 × 23. M 24 {\displaystyle M_{24}} is transitive on octads and on dodecads. The other Mathieu groups occur as stabilizers of one or several elements of W .