
mathue group emm24

samuuha siddhaantam anie piluvabade aadhunika beejaganitam yokka praantamloo, mathue samuuham "m24" anedi kramam yokka chedurumaduru saadhaarana samuuham.
charitra mariyu aastulu.
"m24" anedi 26 chedurumaduru samoohaalalo okati mariyu dheenini parichayam chesaaru. idhi 24 vastuvulapai 5-parivartana kramaparivartana samuuham. shhar multiplier mariyu bayati automorphism group rendoo chinnavi.
mathue samuuhaalanu vividha margallo nirminchavacchu. praarambhamlo, mathue mariyu itarulu vaatini kramaparivartana samuhaluga nirminchaaru. m24 vaastavaaniki unikilo undani, daani generatorlu pratyaamnaaya samuuham a24nii matrame utpatti cheyaledani chudatam kashtam. ernst whit m24nii s (5,8,24) steiner system W24 (whit design) yokka automorphism (cemetry) groupugaa nirminchinappudu eee vishayam spashtam cheyabadindi. m24 anedi eee roopakalpanaloni prati blacknu vaerae blacku mapp chese prastaaranala samuuham. appudu m23 mariyu m22 upa samuuhaalu varusaga oke binduvu mariyu oka jatha binduvula sthireekaranagaa sulabhamgaa nirvachinchabadataayi.
oka kramabaddheekarana samuuhamgaa nirmaanam.
m24 anedi moodu prastaaranala dwara utpatti cheyabadina s24 yokka upa samuuham.
m24 nuu rendu prastaaranala dwara kooda utpatti cheyavachuhu
psl (3,4) nundi m24.
m24 nii psl (3,4) nundi praarambhinchi nirminchavacchu, idhi 4 muulakaalatoe parimita kshetrampai trimiteeya sthalam yokka projective pratyeka sarala samuuham. konnisaarlu m21 anie piluvabade eee samuuham, f4 feeled meeda projective plain meeda panichestundi, idhi W21 anie piluvabade s (2,5,21) vyavastha. daani 21 blaakulanu linelu antaaru. edainaa rendu rekhalu oka binduvu vadda kalustaayi.
m21loe 360 kramamlo 168 saadhaarana upa samuuhaalu mariyu 168 kramamlo 360 saadhaarana upa samuuhaalu unnaayi. pedda projective general leanier group pgl (3,4) loe rendu upa samuuhaalu oke samyoga taragatulanu yerparustaayi, conny m21loe rendu samuuhaalu 3 samyoga taragatulugaa vibhajinchabaddaayi. upa samuuhaalu varusaga 6 kaksyalanu kaligi untaayi, veetini hyperovals anie pilustaaru, mariyu 7 kaksyalanu phano sub plains anie pilustaaru. eee setlu pedda steiner vyavasthala kosam kotta blaakulanu roopondinchadaaniki anumatistaayi. suuchika 3 looni pgl (3,4) loe m21 saadhaaranam. pgl (3,4) anedi f4 (feeled automorphism) loe samyoga muulakaalanu badilee cheyadam dwara prerepinchabadina bahya automorphismnu kaligi untundi. anduvalla pgl (3,4) nuu projective semilinear transformationla pγl (3,4) groupuku vistarinchavacchu, idhi s3 cemetric group dwara m21 yokka split extension. pγl (3,4) anedi m24 yokka garishta upagroopgaa embeddingnu kaligi untundi.
hyperoval kuu 3 paayintlu kaalineer gaa undavu. oka phano sub plain kooda adevidhamgaa tagina pratyeka paristhitulanu santruptiparustundi.
W21 kuu 3 kotta paayintlanu jodinchi, pγl (3,4) loe automorphismsnu anumatinchandi, conny m21loe eee kotta paayintlanu kramabaddheekarinchavaddu. a s (3,6,22) vyavastha W22 anedi 21 panktulalo pratidaaniki kevalam oka kotta binduvunu jodinchadam dwara erpadutundi mariyu kotta blaakulu m21 kinda 56 hyperovals samyogam kaligi untaayi.
oka s (5,8,24) vyavasthalo 759 blaakulu ledhaa actadlu untaayi. W21 yokka prati panktiki mottam 3 kotta paayintlanu jodinchandi, 120 yokka prati setlaloni phano sub plaenlaku vaerae kotta point, mariyu anni hyperovalsku tagina jatala kotta paayintlanu jodinchandi. idhi 210 ashtapadaalu minaha annintikee kaaranamavutundi. migilina ashtapadaalu W21 yokka upasamitulu mariyu jatala rekhala yokka samaroopa vyatyaasaalu. pγl (3,4) samuuhaanni m24kii vistarinchadaaniki aneka margalu unnaayi.
gole code yokka automorphism samuuham.
m24 samuuham kooda binary gole code "w", i yokka kramaparivartana automorphism samuuham.ii., "doubleu" nuu daaniki mapper chese coordinatela kramapallavaala samuuham. sanketapadaalu 24 vastuvula samiti yokka upasamitulaku sahajamgaa anugunamgaa untaayi. (coding siddhaantamlo "binary gole code" anae padam tarachugaa takkuva sambandhita podavu 23 kodnu suuchistumdi, mariyu ikkada upayoginche podavu 24 kodnu "podiginchina binary gole code" anie pilustaaru.) 1kii samaanamaina 8 ledhaa 12 akshaamsaalatoo sanketapadaalaku anugunamgaa unde upasamitulanu varusaga octads ledhaa dodecads antaaru. actadlu anedi oka s (5,8,24) steiner vyavastha yokka blocks mariyu binary gole code anedi steiner vyavastha yokka actadlu vistarinchi unna feeled f2 paie vector space.
saadhaarana upa samuuhaalu m23, m22, m12, mariyu m11 lanu m24 yokka upa samuhaluga nirvachinchavacchu, varusaga oke coordinate yokka stabilizerlu, kramabaddheekarinchina janta coordinatelu, dodecad mariyu oke coordinate thoo kalisi dodecad.
mathue samuuhaalu mariyu pedda conway samuuhaala madhya sahaja sambandham undhi, endukante binary gole code mariyu leach latis rendoo parimaanam 24 yokka pradaesaalaloo untaayi. conway samuuhaalu rakshasudi samuuhamloo kanipistaayi. robbert griss rakshasudilo kanipinche 20 chedurumaduru samuuhaalanu santoshakaramaina kutumbamgaa mariyu mathue samuuhaalanu modati taramgaa suuchistaadu.
polyhedral samaruupatalu.
clain quartic (moodu uparitala jaatula yokka tesslation yokka samaruupatalu) yokka samaruupatala nundi m24 nii nirminchavacchu, idhi psl (2,7), dheenini adanapu kramaparivartana dwara penchavacchu. eee prastaarananu clain quartic yokka tilingtho 56 tribhujaalato praarambhinchi (24 sheershaalato-samuuham panichese 24 paayintlu), taruvaata 2 tribhujaalalo konnintini chaturasraalugaa mariyu 6 tribhujaalalo ashtabhujaluga yerparuchukovadam dwara vivarinchavacchu, adanamgaa prastaarana "chaturasraalu mariyu ashtabhujaalanu vibhajinche asalu tribhujaakaara tailing yokka anchula yokka rendu mugimpu binduvulanu parasparam marchukovadam". tribhujaalaku rangu veyadam dwara dheenini chudavachu-sambandhita tailing sthalaakrutigaa untundi conny rekhaaganitaparamgaa kaadu, mariyu euclidean 3-pradaesamloe chinna cubicbactahedron (idhi 24 sheershaalanu kooda kaligi untundi) gaa (polyhedrally) nimajjanam cheyavachu.
applications.
umbral moonshine siddhaantam anedi k3 uparitalaalu mariyu m24 madhya paakshikamgaa oohaatmaka sambandham.
kanve group koo1, fisher group phai24, mariyu janco group zhe4 okkokkati garishta upa samuuhaalanu kaligi untaayi, ivi 211 samuuham dwara mathue group emm24 yokka podigimpu. (eee podigimpulu annee okela undavu.)
praatinidhyaalu.
 m24 yokka sanklishta akshara pattikanu lekkinchaaru.
mathue group m24 24 pointlapy 5 retlu parivartana kramaparivartana praatinidhyaanni kaligi undhi. sanklishta sankhyalapai sambandhita sarala praatinidhyam anedi chinnapati praatinidhyam mariyu 23-dimensional irreducible praatinidhyam yokka mottam.
m24 rendu rank 3 kramabaddheekarana praatinidhyaalanu kaligi undih okati 276 = 1+44 + 233 jatala paayintla (ledhaa dwadlu) meeda stabilizer m22.2 thoo, mariyu okati 1288 = 1+495 + 792 dwadlalo, stabilizer m12.2 thoo.
daani 1-dimensional fixed subspace dwara prastaarana praatinidhyam yokka 24-dimensional leanier praatinidhyam yokka coashant 23-dimensional praatinidhyaanni istundi, idhi 2 ledhaa 3 kaani lakshanaala yokka yea feeled meeda ayina irreducible, mariyu atuvanti feeldlapy atichinna nammakamaina praatinidhyaanni istundi.
24 dimensional praatinidhyaanni tagginchadam mode 2 anedi f paie charyanu istundi. idhi dimension 1,12 (gole code) mariyu 23 yokka asthira upasthaanaalanu kaligi undhi. upavibhajanalu 2 muulakaalatoe feeled meeda dimension 11 yokka rendu irreducible praatinidhyaalanu istayi.
garishta upa samuuhaalu.
 m24" yokka garishta upa samuuhaala yokka 9 samyoga taragatulanu kanugonnaru. 24 pointlapy combinatorial dayta paramgaa 9 taragatulanu vivaristuu, phalitaaniki oka chinna rujuvunu ichchaaru upa samuuhaalu krinda vivarinchina vidhamgaa oka point, dwad, octaud, dum, sextet, triad, trayam, projective line ledhaa actornu parishkaristaayi. m24 akshara pattikalu (modata lekkinchinavi) mariyu aaa samayamlo telisina 8 garishta upa samuuhaalanu icchaaru.
m24loe 13 isomorphism rakaalaina abelian kaani saadhaarana upa samuuhaalu unnaih a5 yokka aidu taragatulu, psl (3,2) yokka naalugu taragatulu, a6 yokka rendu taragatulu, psl (2,11) yokka rendu taragatulu, a7 yokka okkokka taragati, psl (2,23), m11, psl (3,4), a8, m12, m22, m23 mariyu m24. a6 kooda sextet upasamuuhamloo upabhaagamgaa krinda perkonabadindi.
mathue samuuham 3 kakshyalato sthira sthalamlo 2048 = 1+759 + 1288 paayintla maadyulo panichestundi, mariyu 4096 = 1+24 + 276+2024 + 1771 paayintla vadda 5 kakshyalato kokod yokka paayintlu, mariyu code ledhaa kokod yokka chinna-kaani binduvunu sthirapariche upa samuuhaalu garishta upa samuuhaala 9 taragatulalo 6nii istayi. 
garishta upa samuuhaala yokka 9 taragatulu eee krindi vidhamgaa unnaih
point upa samuuham.
oka binduvunu nirnayinche upa samuuham m23, arder 10200960.
duvad upa samuuham.
oka dvayam anedi oka jatha paayintlu. oka dvayamnu sarichese upa samuuham
m22:2, arder 887040,2 mariyu 22 kakshyalato.
octaud upa samuuham.
gole code ledhaa steiner vyavastha yokka 759 (= 3·11·23) octadlalo okadaanni nirnayinche upa samuuham octaud group. 
24: a8, kramam 322560, parimaanam 8 mariyu 16 kakshyalato. leanier group gl (4,2) pratyaamnaaya group a8kii asaadhaaranamaina isomorphism kaligi untundi. octaud yokka pointwise stabilizer o anedi arder 16, ghaataankam 2 yokka abelian group, dheeni prati involution octaud velupala mottam 16 paayintlanu kadilistundi. octaud yokka sthireekarana anedi a8 dwara o yokka vibhajana podigimpu. 
dum upa samuuham.
dume anedi gole kodlo paripuurakaramaina dodecadla (12 paayintla setlu) jatha. oka dvayamnu sarichese upa samuuham
m12:2, arder 190080, parivartanaseela mariyu dosharahitam. eee upasamuuhaanni frobenius kanugonnadu. 
upa samuuham m12, m12 yokka bayati automorphismnu pratibimbistuu, 12 yokka 2 setlapai bhinnamgaa panichestundi.
sextet upa samuuham.
26: (3. s6), arder 138240: "sextet group"
oka tetraadnu pariganinchandi, steiner vyavasthalo 4 paayintla samiti W24. oka octaud migilina 20 nundi aidava point empika dwara nirnayinchabadutundi. 5 actadlu saadhyame. anduvalla edainaa tetrad oka vibajananu 6 tetradluga nirnayistundi, dheenini sextet anie pilustaaru, dheeni sthireekarananu m24loe sextet group anie pilustaaru.
tetradla mottam sankhya 24 * 23 * 22 * 21/4! = 23 * 22 * 21. daanini 6thoo bhagiste 23 * 11 * 7 = 1771 anae aksharaala sankhya vastundi. inca, oka aksharaala samuuham anedi 6va kramam yokka pushpaguchcham yokka upa samuuham!*(4!) 6, dheeni ekaika pradhana bhajakalu 2,3 mariyu 5. ippudu manaku pradhana bhajakalu telusu |m24|. tadupari vislaeshanha sextet samuuham yokka kramaanni nirnayistundi, anduvalla |m24|.
24 paayintlanu 6-bye-4 shrenilo amarchadam soukaryavantamgaa untundih
antekakunda, varusalanu sankhya cheyadaaniki f4 feeled yokka muulakaalanu upayoginchadam soukaryavantamgaa untundih 0,1, u, u2.
sextet samuuhamloo kramam 64 yokka saadhaarana abelian upa samuuham h untundi, idhi hexacodeku isomorphic, podavu 6 yokka vector space mariyu f4 paie parimaanam 3. h looni sunna kaani moolakam niluvu varusalo 4 ledhaa 6 lopala double transpositionlanu chestundi. daani charyanu varusa sankhyalaku vector coordinatesnu jodinchadam anie bhaavinchavachchu.
sextet samuuham anedi 3. s6 (kaandam podigimpu) dwara h yokka vibhajana podigimpu. ikkada mathue samoohaalalo oka udaaharana undhi, ikkada oka saadhaarana samuuham (a6) oka upasamuuham kaadu, upabhaagamu. 3.s6 anedi r = (bcd) (fh) (jkl) (nop) (rst) (vwx) dwara utpatti cheyabadina upasamuuham yokka m24loe saadhaaraneekarana, dheenini yuu2 dwara varusa sankhyala gunanamgaa bhaavinchavachchu. upasamuuham 3. a6 anedi daani kendrakamgaa untundi. 3. a6 yokka generatorlu
niluvu varusala yokka besi kramaparivartana, (cd) (gh) (kl) (op) (q) (rv) (sx) (tw), appudu 3. s6 nuu utpatti chestundi.
samuuham 3. a6 anedi sl (3,4) yokka upasamuuhaaniki isomorphic, dheeni chitram psl (3,4) loe hyperoval samuuhamgaa paina perkonabadindi.
opplet mogeelo rangulo sextetlanu pradarsinche function untundi.
trayam upa samuuham.
oka trayam anedi 3 paayintla samiti. trayamnu nirnayinche upa samuuham 
psl (3,4): s3, arder 120960, parimaanam 3 mariyu 21 kakshyalato.
trayam upa samuuham.
trayam anedi gole code yokka 3 vidadeeyabadina ashtapadaala samiti. trayamnu sarichese upa samuuham trayam samuuham
26: (psl (2,7) x s3), arder 64512, parivartanaseela mariyu dosharahitam.
projective line upa samuuham.
24 paayintla meeda projective line structurenu fixes chese upa samuuham
psl (2,23), arder 6072, dheeni charya rettimpu parivartanaseelamgaa untundi. eee upasamuuhaanni mathue gamaninchaaru.
actern upa samuuham.
actorn antey 24 paayintlanu 3 yokka 8 blaculuga vibhajinchadam. actornu nirnayinche upa samuuham 
actern group anedi 168va kramamlo psl (2,7) kuu isomorphic, saralamaina, parivartanaseelamaina mariyu dosharahitamainadi.
idhi m24 yokka chivari garishta upa samuuham.
samyoga taragatulu.
26 samyoga taragatulu unnaayi. chakra akaaraalu annee samatulyamgaa untaayi, avi samyoga taragatini batti konni puurnaamkam "n" kosam "k" chakraala podavu "n"/"k" chakralaku maarutuu untaayi.

