THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED
We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; that is, a free...
Gespeichert in:
| Veröffentlicht in: | Transactions of the American Mathematical Society 2010-03, Vol.362 (3), p.1351-1369 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 1369 |
|---|---|
| container_issue | 3 |
| container_start_page | 1351 |
| container_title | Transactions of the American Mathematical Society |
| container_volume | 362 |
| creator | BRAY, JOHN N. CURTIS, ROBERT T. |
| description | We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway's elementsοT and the 24— dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under of the integral vectors in R24. |
| format | Article |
| fullrecord | <record><control><sourceid>jstor</sourceid><recordid>TN_cdi_jstor_primary_40590871</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>40590871</jstor_id><sourcerecordid>40590871</sourcerecordid><originalsourceid>FETCH-LOGICAL-j107t-c801557f06b7b61f756e84a53dc346f8f96efd2c1faeec25198f945a5a30c9fb3</originalsourceid><addsrcrecordid>eNotjN1qwjAYhnPgwJ95CUJuoONL839YYrSFYodGx44kjQlYFEfrya5l97RrmmM7enmeB94RmgBAnmnN5BhNh6F7IDAlJki40uLaWlPiunCuMhZ_f-Fis8S_wTSbt-Idr7fN_hW_NHhrD9Wucnb5jJ6Svwxx_r8ztF9ZZ8qsbtaVKeqsIyDvWVBAOJcJRCtbQZLkIirmOT0FykRSSYuYTnkgyccYck70QzHuuacQdGrpDC3-frvhfuuPH_356vvPIwOuQUlCfwAxIDme</addsrcrecordid><sourcetype>Publisher</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED</title><source>American Mathematical Society Publications (Freely Accessible)</source><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><source>American Mathematical Society Journals</source><source>EZB-FREE-00999 freely available EZB journals</source><creator>BRAY, JOHN N. ; CURTIS, ROBERT T.</creator><creatorcontrib>BRAY, JOHN N. ; CURTIS, ROBERT T.</creatorcontrib><description>We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway's elementsοT and the 24— dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under of the integral vectors in R24.</description><identifier>ISSN: 0002-9947</identifier><language>eng</language><publisher>American Mathematical Society</publisher><subject>Bricks ; Combinatorics ; Highway interchanges ; Mathematical lattices ; Mathematical permutation ; Mathematical sets ; Mathematical vectors ; Sextets ; Symmetry</subject><ispartof>Transactions of the American Mathematical Society, 2010-03, Vol.362 (3), p.1351-1369</ispartof><rights>2010 American Mathematical Society</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/40590871$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/40590871$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,803,832,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>BRAY, JOHN N.</creatorcontrib><creatorcontrib>CURTIS, ROBERT T.</creatorcontrib><title>THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED</title><title>Transactions of the American Mathematical Society</title><description>We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway's elementsοT and the 24— dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under of the integral vectors in R24.</description><subject>Bricks</subject><subject>Combinatorics</subject><subject>Highway interchanges</subject><subject>Mathematical lattices</subject><subject>Mathematical permutation</subject><subject>Mathematical sets</subject><subject>Mathematical vectors</subject><subject>Sextets</subject><subject>Symmetry</subject><issn>0002-9947</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNotjN1qwjAYhnPgwJ95CUJuoONL839YYrSFYodGx44kjQlYFEfrya5l97RrmmM7enmeB94RmgBAnmnN5BhNh6F7IDAlJki40uLaWlPiunCuMhZ_f-Fis8S_wTSbt-Idr7fN_hW_NHhrD9Wucnb5jJ6Svwxx_r8ztF9ZZ8qsbtaVKeqsIyDvWVBAOJcJRCtbQZLkIirmOT0FykRSSYuYTnkgyccYck70QzHuuacQdGrpDC3-frvhfuuPH_356vvPIwOuQUlCfwAxIDme</recordid><startdate>20100301</startdate><enddate>20100301</enddate><creator>BRAY, JOHN N.</creator><creator>CURTIS, ROBERT T.</creator><general>American Mathematical Society</general><scope/></search><sort><creationdate>20100301</creationdate><title>THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED</title><author>BRAY, JOHN N. ; CURTIS, ROBERT T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-j107t-c801557f06b7b61f756e84a53dc346f8f96efd2c1faeec25198f945a5a30c9fb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Bricks</topic><topic>Combinatorics</topic><topic>Highway interchanges</topic><topic>Mathematical lattices</topic><topic>Mathematical permutation</topic><topic>Mathematical sets</topic><topic>Mathematical vectors</topic><topic>Sextets</topic><topic>Symmetry</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>BRAY, JOHN N.</creatorcontrib><creatorcontrib>CURTIS, ROBERT T.</creatorcontrib><jtitle>Transactions of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>BRAY, JOHN N.</au><au>CURTIS, ROBERT T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED</atitle><jtitle>Transactions of the American Mathematical Society</jtitle><date>2010-03-01</date><risdate>2010</risdate><volume>362</volume><issue>3</issue><spage>1351</spage><epage>1369</epage><pages>1351-1369</pages><issn>0002-9947</issn><abstract>We give a new, concise definition of the Conway group ·O as follows.The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $(_4^{24} )$tetrads. We use this action to define a progenitor P of shape $2*(_4^{24} )$M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway's elementsοT and the 24— dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under of the integral vectors in R24.</abstract><pub>American Mathematical Society</pub><tpages>19</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0002-9947 |
| ispartof | Transactions of the American Mathematical Society, 2010-03, Vol.362 (3), p.1351-1369 |
| issn | 0002-9947 |
| language | eng |
| recordid | cdi_jstor_primary_40590871 |
| source | American Mathematical Society Publications (Freely Accessible); JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; American Mathematical Society Journals; EZB-FREE-00999 freely available EZB journals |
| subjects | Bricks Combinatorics Highway interchanges Mathematical lattices Mathematical permutation Mathematical sets Mathematical vectors Sextets Symmetry |
| title | THE LEECH LATTICE Λ AND THE CONWAY GROUP .O REVISITED |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-03T22%3A47%3A01IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=THE%20LEECH%20LATTICE%20%CE%9B%20AND%20THE%20CONWAY%20GROUP%20.O%20REVISITED&rft.jtitle=Transactions%20of%20the%20American%20Mathematical%20Society&rft.au=BRAY,%20JOHN%20N.&rft.date=2010-03-01&rft.volume=362&rft.issue=3&rft.spage=1351&rft.epage=1369&rft.pages=1351-1369&rft.issn=0002-9947&rft_id=info:doi/&rft_dat=%3Cjstor%3E40590871%3C/jstor%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rft_jstor_id=40590871&rfr_iscdi=true |