
oh 'naan-scott pramey

ganit mein, oh 'naan-scott pramey kramparivartan samooh siddhant key sabse prabhaavshaali prameyon mein seey ack haye; seemit saral samoohon kaa vargeekaran hii ise itna upayogi banaata haye. mool roop seey pramey samamit samooh key adhikatam upsamuhon key baare mein thaa. yeh 1979 mein seemit samoohon parr saanta cruze sammelan key liye likhe gaye leonard scott key ack pepper key parishisht key roop mein dikhaai diyaa, jisamen ack futnote thaa kii michael oh 'naan ney swatantra roop seey vahee parinam saabit kiya thaa. michael ashbacker aur scott ney baad mein pramey key kathan kaa ack sahee sanskaran diyaa.
pramey bataata haye kii samamit samooh kaa ack adhikatam upsamooh (ω), jahan .ω. = "aine", nimnalikhit mein seey ack haih 
landan ganitiya samaaj key bulletin key liye likhe gaye ack sarvekshan patra mein, 
peater jey. aisaa lagata haye kii cameron ney sabse pahale yeh pahachaana haye kii oh 'naan-scott pramey mein vaastavik shakti seemit adim samoohon koo vibhinn prakaron mein vibhajit karane kii kshamata mein haye.
ack swa-nihit pramaan key saath pramey kaa ack poorna sanskaran m dwara diyaa gayaa thaa.doubleu. libeck, cheryl pregor aur zaan saxel. pramey abb kramparivartan samoohon parr pathyapustakon kaa ack maanak hissa haye.
oh 'naan-scott prakaar.
seemit adim kramparivartan samoohon key aathh oh 'naan-scott prakaar iss prakaar hainh
haa (ack abelian samooh kaa holomorph): yeye adim samooh hain joe kuchh prime "pee" aur sakaratmak purnank "dee" ≤1 key liye affine samanya raikhik samooh aeye. jee. ell. ("dee", "pee") key upsamooh hain. aise samooh "jee" key liye, isamen sabhi anuvaadon kaa upsamooh honah chaahiye, aur shoonya sadish key "jee" mein sthirikaran g0 koo gla ("dee, pee") kaa ack irreducible upsamooh honah chaahiye. haa prakaar key adim samoohon kii visheshta ack advitiya nyuntam samanya upsamooh haye joe praathamik abelian haye aur niyamit roop seey kaary kartaa haye.
hs (ack saral samooh kaa holomorph): "tii" koo ack seemit gair-labelian saral samooh honey dein. taba "m" = "t" × "t" ω = "t" bataa "t" ("t" 1, "t" 2) = "t" 1 − 1 "tt" 2 parr kaary kartaa haye. abb "m" mein dou nyuntam samanya upsamooh "n" 1 "," n "2" hain, pratyek samroopata seey "t" haye aur pratyek niyamit roop seey ω parr kaary kartaa haye, ack daaen gunaa seey aur ack baaen gunaa seey. m" kii kriya adim haye aur yadi humm "α" = 1 "t" lete hain tou hamaare paas "m" α "= {(" t "," t ") haye.."t" "bul" "t" "}, jisamen ω parr inn (" "t" ") shamil haye. vaastav mein "tii" kaa koee bhee swachaalan ω parr kaary karega. hs prakaar kaa ack adim samooh taba koee bhee samooh "jee" hotaa haye joe "aim" tii "hotaa haye.inn ("tii") ≤ "jee" ≤ "tii".auto ("tii"). aise sabhi samoohon mein nyuntam samanya upsamooh key roop mein "aine" 1 aur "aine" 2 hotey hain.
aich. see. (ack yaugik samooh kaa holomorph): maan leejiye kii "tii" ack gair-labelian saral samooh haye aur kuchh purnank "key" ≤2 key liye "aine" 1 "aine" 2 "tii" key "maan leejiye. maan leejiye kii ω =" tii "key". taba "m" = "n" 1 × "n" 2 "x" ("n" 1, "n" 2) key madhyam seey ω parr pargamansheel roop seey kaary kartaa haye = "n" 1 − 1 "xn" 2 sabhi "x" ω "," n "1 ω" n "1", n "2 ω" n "2 key liye. jaise kii hs mamle mein, hamaare paas" m "̃" t "k" haye.inn ("tii" key ") aur" tii "key" kaa koee bhee swarupikaran bhee ω parr kaary kartaa haye. aich. see. prakaar kaa ack adim samooh ack samooh "jee" haye joe "aim" ≤ "jee" ≤ "tii" key "haye.aut ("tii" key ") aur" jee "aut (" tii "key") = aut ("tii") wr "aiss" key "key" key "key" saral pratyaksh karakon key sett parr paargamanik roop seey kaary karate hain. aise kissy bhee "jee" mein dou nyuntam samanya upsamooh hotey hain, pratyek samroop seey "tii" key "aur niyamit.
aich. see. prakaar kaa ack samooh ack utpaad sanrachana koo sanrakshit kartaa haye jahan δ = "t" aur "g" ≤ "h" wr "s" k "jahan" h "δ parr prakaar hs kaa ack adim samooh haye.
twa (twisted pushpamala): yahaan "jee" mein ack advitiya nyuntam samanya upsamooh "aine" aur "aine" tii "key" kuchh seemit gair-labelian saral samooh "tii" key liye haye aur "aine" niyamit roop seey ω parr kaary kartaa haye. aise samoohon kaa nirmaan mudey hue pushp maalaa utpaadon key roop mein kiya jaa sakata haye aur isaliye label twa. adimata praapt karane key liye aavashyak sharton kaa arth haye kii "k" ≤6 tou iss tarah key adim samooh kii sabse chhoti degree 606 haye.
jaisa kii (lagbhag saral): yahaan "jee" "tii" aur "tii" ("tii") key beach sthit ack samooh haye, arthaat, "jee" lagbhag saral samooh haye aur isaliye naam haye. hamein iss baare mein kuchh naheen bataaya jaataa haye kii kaarravaai kyaa haye, isake alawa kii yeh adim haye. iss prakaar key vishleshan key liye lagbhag saral samoohon key sambhaavit adim karyon key baare mein jaanane kii aavashyakta hoti haye, joe lagbhag saral samoohon key adhikatam upsamuhon koo jaanane key barabar haye.
aiss. dee. (saral vikarn): kuchh gair-labelian saral samooh "tii" aur purnank "key" ≤2 key liye "aine" = "tii" key "aur" aich "= {(" tii ")..., tii "). "t" "bul" "t" "} ≤" "n" ". phir "n" sahee gunan dwara "n" mein "h" key sahee coasuts key sett ω parr kaary kartaa haye. humm {("t" 1 leey sakate hain.., tii" "key"-1,1) . "t" "i" "ω" "t" "koo" "n" "mein" "h" "key liye coaset pratinidhiyon kaa ack samooh honah chaahiye aur isaliye humm ω koo" "t" "k" "−1 key saath pehchaan sakate hain. abb (" "s" "1." "s" "1".., s" "k") ω "" n "pratinidhi (" "t" "1 key saath coaset letaa haye.., t" "k"-1,1) coaset "h" ("t" 1 "s" 1.., t" "k" "−1" "s" "k" −1 "," s "" k ") =" "h" ("" s "" k "−1" "t" "k" "s" 1. ., s" "k" "−1" "t" "k" −1 "" s "" k "−1). samooh "s" k "pravishtiyon koo krambaddh karke" n "key swachaalan koo prerit kartaa haye aur upsamooh" h "koo thheek kartaa haye aur isaliye sett parr kaary kartaa haye. yeh bhee dhyaan dein kii "h" inn ("t") koo prerit karke ω parr kaary kartaa haye aur vaastav mein "t" kaa koee bhee automorphism σ pratinidhi ("t" 1 key saath coaset koo laker ω parr kaary kartaa haye.., t" "k" −1) pratinidhi ("" t "1σ key saath coaset key liye..., t" "k" −1σ, 1). iss prakaar humm ack samooh "doubleu" = "aine" praapt karate hain.(baahar ("tii") × "sk") ≤ simm (ω). aiss. dee. prakaar kaa ack adim samooh ack samooh "jee" ≤ "doubleu" haye joe "aine" "jee" aur "jee" "aine" key "key" saral pratyaksh karakon parr "aiss" "key" key ack adim upsamooh koo prerit kartaa haye.
cd (yaugik vikarn): yahaan ω = δ "k" aur "g" ≤ "h" wr "sk" jahan "h" nyuntam samanya upsamooh "tl" key saath δ parr prakaar sd kaa ack adim samooh haye. isake alawa, "n" = "t" "kl" "g" kaa ack nyuntam samanya upsamooh haye aur "g" "s" "k" key ack paargamanik upsamooh koo prerit kartaa haye.
paa (utpaad kriya): yahaan ω = δ "k" aur "g" ≤ "h" wr "s" k "jahan" h "δ parr ack adim lagbhag saral samooh haye jisamen sokle" t "haye. iss prakaar "g" kii ω parr ack utpaad kriya haye. isake alawa, "n" = "t" k "" g "aur" g "" n key "k" saral pratyaksh karakon parr apani kriya mein "Sk" key ack sankramansheel upsamooh koo prerit karate hain."
kuchh lekhak vibhinn prakaar key vibhajan kaa upayog karate hain. sabse aam haye kii hs aur sd prakaron koo ack "vikarn prakaar" key roop mein aur hc, cd aur pee. aeye. prakaron koo ack "utpaad kaarravaai prakaar" key roop mein shamil kiya jaae."pregor ney baad mein oh 'naan-scott pramey koo quassipremetive samoohon (wafadar karyon vaale samooh jaise kii pratyek gair-tuchchh samanya upsamooh key liye pratibandh sankramansheel haye) key liye saamaanyikrit kiya.

