MathieuGroupM22

MathieuGroupM22[]

represents the sporadic simple Mathieu group .

Details

  • By default, MathieuGroupM22[] is represented as a permutation group acting on points {1,…,22}.

Background & Context

  • MathieuGroupM22[] represents the Mathieu group , which is a group of order . It is one of the 26 sporadic simple groups of finite order. The default representation of MathieuGroupM22 is as a permutation group on the points having two generators.
  • The Mathieu group is the fourth smallest of the sporadic finite simple groups. It was discovered (along with the other four Mathieu groups MathieuGroupM11, MathieuGroupM12, MathieuGroupM23 and MathieuGroupM24) by mathematician Émile Léonard Mathieu in the late 1800s, making these groups tied for first in chronological order of discovery among sporadic groups. MathieuGroupM22 is 3-transitive in the sense that there exists at least one group element mapping any unique 3-tuple of elements of MathieuGroupM22 to any other unique 3-tuple therein. In addition to its permutation representation, can be defined in terms of generators and relations as and has a 3-transitive representation on , the point stabilizer of which is the projective special linear group . Along with the other sporadic simple groups, the Mathieu groups played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to MathieuGroupM22[], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the Mathieu group are available via FiniteGroupData[{"Mathieu",22},"prop"].
  • MathieuGroupM22 is related to a number of other symbols. Along with MathieuGroupM11, MathieuGroupM12, MathieuGroupM23 and MathieuGroupM24, MathieuGroupM22 is one of five groups cumulatively referred to as the so-called "first generation" of sporadic finite simple groups. It is also one of 20 so-called "happy" sporadic groups, which all appear as a subquotient of the monster group.

Examples

Basic Examples  (1)

Order of the group :

Generators of a permutation representation of the group :

Wolfram Research (2010), MathieuGroupM22, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuGroupM22.html.

Text

Wolfram Research (2010), MathieuGroupM22, Wolfram Language function, https://reference.wolfram.com/language/ref/MathieuGroupM22.html.

CMS

Wolfram Language. 2010. "MathieuGroupM22." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MathieuGroupM22.html.

APA

Wolfram Language. (2010). MathieuGroupM22. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MathieuGroupM22.html

BibTeX

@misc{reference.wolfram_2024_mathieugroupm22, author="Wolfram Research", title="{MathieuGroupM22}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MathieuGroupM22.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_mathieugroupm22, organization={Wolfram Research}, title={MathieuGroupM22}, year={2010}, url={https://reference.wolfram.com/language/ref/MathieuGroupM22.html}, note=[Accessed: 22-November-2024 ]}