LyonsGroupLy

LyonsGroupLy[]

represents the sporadic simple Lyons group .

Details

  • No permutation representation is implemented for LyonsGroupLy[].

Background & Context

  • LyonsGroupLy[] represents the Lyons group , a group of order TemplateBox[{2, 8}, Superscript].TemplateBox[{3, 7}, Superscript].TemplateBox[{5, 6}, Superscript].7.11.31.37.61. It is one of the 26 sporadic simple groups of finite order. The Lyons group is also referred to as the Lyons–Sims group and denoted .
  • The Lyons group is the eighth largest of the sporadic finite simple groups. It was discovered by mathematician Richard Lyons and constructed explicitly by Charles Sims in the early 1970s. LyonsGroupLy was first described as the unique possible group having an involution with centralizer isomorphic to the nontrivial central extension of AlternatingGroup[11] by CyclicGroup[2]. The Lyons group has a number of different representations, including a pair of rank-5 permutation representations on and symbols (the prior of which is faithful), as well as a modular representation of dimension 111 over the field with five elements. Along with the other sporadic simple groups, played a foundational role in the monumental (and complete) classification of finite simple groups.
  • The usual group theoretic functions may be applied to LyonsGroupLy[], including GroupOrder, GroupGenerators, GroupElements and so on. However, while LyonsGroupLy[] is a permutation group, due its large order, an explicit permutation representation is impractical for direct implementation. As a result, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the Lyons group are available via FiniteGroupData["Lyons","prop"].
  • LyonsGroupLy is related to a number of other symbols. Along with JankoGroupJ1, JankoGroupJ3, JankoGroupJ4, ONanGroupON and RudvalisGroupRu, LyonsGroupLy is one of six sporadic simple groups referred to as "pariahs" as a consequence of their failure to occur as subquotients of the monster group.

Examples

Basic Examples  (1)

Order of the Lyons group :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_lyonsgrouply, author="Wolfram Research", title="{LyonsGroupLy}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/LyonsGroupLy.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_lyonsgrouply, organization={Wolfram Research}, title={LyonsGroupLy}, year={2010}, url={https://reference.wolfram.com/language/ref/LyonsGroupLy.html}, note=[Accessed: 22-December-2024 ]}