{"id":6592,"date":"2024-09-28T16:45:49","date_gmt":"2024-09-28T23:45:49","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=6592"},"modified":"2026-01-11T17:23:30","modified_gmt":"2026-01-12T00:23:30","slug":"group-theory-in-a-nutshell-taming-the-monster","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2024\/09\/28\/group-theory-in-a-nutshell-taming-the-monster\/","title":{"rendered":"Group Theory in a Nutshell: Taming the Monster"},"content":{"rendered":"\n

For a paper with much of the content below, please see: theoryofeverything.org\/TOE\/JGM\/E8_and_H4_in_QM_and_QC.pdf<\/a>, and associated Mathematica notebook here.<\/a><\/p>\n\n\n\n

<\/p>\n\n\n

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Y555<\/sub> Hasse diagram<\/figcaption><\/figure>\n<\/div>\n\n\n

The wreath product of the Monster group with Z2 (M \u2240 Z2) is the BiMonster, a quotient of the Y555<\/sub> Dynkin diagram which contains six E8(Y124<\/sub>) Dynkin diagrams and the 23rd Niemeier lattice E83<\/sup> root system. This can be reduced to Y444<\/sub> by applying the spider relation to E83<\/sup> .<\/p>\n\n\n

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Hasse diagram of sporadic groups in their Monster group subquotient relationships. <\/figcaption><\/figure>\n<\/div>\n\n\n

Happy Family generations:
Red=Mathieu groups (1st)
Green=Leech lattice groups (2nd)
Blue=other Monster subquotients (3rd)
Black=Pariahs<\/p>\n\n\n

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Table of sporadic group orders (with Tits group).<\/figcaption><\/figure>\n<\/div>\n\n
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Niemeier lattices with group structure, Dynkin diagram, Coxeter number, and group orders.<\/figcaption><\/figure>\n<\/div>\n\n
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Kneser neighborhood graph of Niemeier lattices with associated Mathematica Notebook here<\/a>.<\/figcaption><\/figure>\n<\/div>\n\n\n

Each color coded node represents one of the 24 Niemeier lattices, and the lines joining them represent the 24-dimensional odd unimodular lattices with no norm 1 vectors. The Coxeter number of the Niemeier lattice is to the left. The red node index number indicates the row in the table above.<\/p>\n\n\n

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Table of sublattices of Conway groups (Green nodes in the Monster Group sub-quotients graph at the top of the page)<\/figcaption><\/figure>\n<\/div>\n\n
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The Freudenthal magic square<\/figcaption><\/figure>\n<\/div>\n\n\n

\u201dThe Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that \u201dthe exceptional Lie groups all exist because of the octonions: G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space, see prehomogeneous vector space.\u201d<\/p>\n\n\n

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E8 Subgroup Tree with associated Mathematica Notebook here<\/a>. <\/figcaption><\/figure>\n<\/div>\n\n\n

This E8SubgroupTree.nb<\/a> Mathematica Notebook (140Mb), viewable using the free viewable Player<\/a>, which has interactive Tooltips on group nodes (with group data and Hasse diagram) and Tooltips on the subgroup edges (with its individual maximal embedding chart). Also in the notebook is one cell with a table of all of the above data plus the Dynkin diagram based maximal embedding charts and an example 2D or 3D polytope projection.<\/p>\n","protected":false},"excerpt":{"rendered":"

For a paper with much of the content below, please see: theoryofeverything.org\/TOE\/JGM\/E8_and_H4_in_QM_and_QC.pdf, and associated Mathematica notebook here. The wreath product of the Monster group with Z2 (M \u2240 Z2) is the BiMonster, a quotient of the Y555 Dynkin diagram which contains six E8(Y124) Dynkin diagrams and the 23rd Niemeier lattice E83 root system. This can … Continue reading Group Theory in a Nutshell: Taming the Monster<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[8,4,34,14,16],"class_list":["post-6592","post","type-post","status-publish","format-standard","hentry","category-physics","tag-3d","tag-e8","tag-h4","tag-math","tag-physics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6592","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=6592"}],"version-history":[{"count":48,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6592\/revisions"}],"predecessor-version":[{"id":7707,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/6592\/revisions\/7707"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=6592"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=6592"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=6592"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}