Osnovy teorii Galua.

_____ __ _ ~ 7. Pa3pewuubte epynnbi__ 4
wKeHHg. ECAH MOxiy~ib Ii xonyc~aCT n e p io o6 pa 3 11 e K0 PH H ('ITO timeeT mecTo
B3 Tex cay'1agx, ecail fl eCTb HilH CTe11CI~b Rpoc~oro 'IHcjla, WIIH YJABoeRHaq CTeneHb
npoCToro 'IHc.ia, 11au1 qiicjio 4), TO mb! HoJayqIM LUiia~CauCcYlO rpyflny; B npOTHBHOM
cniyqae rpyiiua 6yaleT a,,)e je B or1, HO He IIHKjniMiecKoM-.
10. HYCmb n = 27 = 3- B33HMHo npOCTblC C 27 BuA'eTbi 1n0 mo~yiIo, 27 TaKObbl:
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26. '~ticao 2 eCmb [epBoO6pa3Hlbfit KopeHb. jjei'CT8HTe lbH, BO303maq 2 ii HocJIeaoBaTeabHbie CTerieHH, Mb!nlolyIHm Bce K-naccbl:
21~ 23, 212-19, 218 =11II 214 =-2 2, 215~ 1 7, 2'~7, 217 14, 2i81 I (mod 27).
[-pynna BIA'IeTOB HO moJiynHo 27 eCTb C.TCILOBaTealbHO LUIKJJMi i.cias r-pynnia HlopuI~ica 18. Bce ee 3J!emelTb] MOFYT 6bl~m npeaCTaB cufbl, Kaic cenenu 2.
29. n 50=2-5 ~.3 eCmb uepBoo6pa3Hb~lf KOpeiib. B03Baoq 3 II nocJaeOBaTe~lbHble Cenefll, 6yhqeM HIMC Tb caeapyoui~e Bbl'IeTbi:
3, 9, 27, 31, 43, 29, 37, ii, 33, 49, 47, 41, 23, 19, 7, 21, 13, 39, 17.
rpynna BbI'IeTOB eCTb L1MKuiHqe.CKaff rpytina 11opHtK 20.
3'. n = 60 = 22-.3.5. Fpynna BbI'IeTOB MO)KCT 6bwTb upeuIC~anwieiia KaKc Hp~moe
l~pHRee~ oAHoR rpynr~ib nIopIIJf 4 li RByx rpyrun HOPRnalca 2:1
=749+143 + I) (11 A- 1) (19 +1)
('I epToVl Caepxy Mbl oTmeIaem,, 'ITO 3ACCb u~caa AtOJ1>Hbi paCCMaTpHBaTbCSl, Kai< gaemeHTbi rpynnbi, TaK 'ITO 31aK + He J1aCT Ham iipana ix cKaJl3JbIBaTb).
40, nl =32 =25, 14HCAO5 HBJIqeTC5! TaK MI3bIBaembiM nIon ynepBoo6pa3 HblM
K 0p H eM, TatK gaK ero CTenHtHM aaIOT n1ojiOBHIHY BCeX BbI'IeTOB. BOT STil ubi'ieTbi:
5, 2 5, 2 9, 1 7, 2 1, 9, 1 3, 1.
MeRHsu rrpi HH4X 311K. ii Hpt6aBJI5!5! no 32 ('ITo6bI caegjia~b B1A'IqTig 1`10O)KHTeAbHbltdH), noqyqHMM
27, 7, 3, 15, ti, 23, 19, 31.
TaKHM o6pa3oM rpyinia BbW,-TOB eCTb ripsioe npOu3B~efflee tjLHKajIIIecKoH'
rpyrnnbi nopsiAia 8 iiIt MjK u'IeCKoft rtoynniw nopniic1a, 2:
2 I(+ 5~25 4-29 +17 +21~9~ + 3) (1+- I).
Yniipa KH eHliie 20. Cywmec~ye'r Au3a Tuna1 a6enlenbix rpynn nopflua p2, r~e
V ripocToe tIHCAlO: LLHKJIUIecK~e 1r-pynlbi it tiplmbie npoH3B3eaeHHl ixHKnR'qecgHX
rpynrn nop-tzA~a p. JaoKa3aTb, 'ITO rpynnibi repB0oro TMH3 coaepwaT TOJlbKO OJLHY 1OJZrpyriny nopsiiwa P, B TO BemqI gai( y r'pyflilbl lTOpor'o TtiUa IX BceUo p~ + 1
O6o6tU.Hu 3TOT pe3yJlbTaT Ha r'pyuni~h 6onee o6umero Tuna.
~ 7 Paz~pemrmbie rpyrnaw.
f plpO3BOjAHasi rpynna. BBeaeM B paCCMoTpeffite no.upynrny, HaabIBaemyio HV0M3 BOA1HOR rpyn notlHn Wi KommyTaHTOM 3a~taHHOfl.rpynnlb. TaK Hu3b!Bae-rcs rpyrina, rnojny'aemasi OT K-OMI1O3MLLHH KO0M MYTafro p OB, T. e. aa e m eUiTB wiaTa ABA,-1 B -1 ' rae A i-i B - aieMe~rbl
3aLaafHOR rpynnfbl. OTHOCMTe.bHol np~OH3BOJ1HbIll rpyrin HmMeIT meCTO cvieAyioti~e Teopembi:
T eop e ma 32. HpoV133BO uan rpyiuia eCTb HopmaJ~bHW~ AleunHTejib
3aJaaHHofl. Bonee Toro: npll0M3BOJaFl rpynna eCTb Hopma~lbHblfl aenlHTeAb B3C5IKOt rpyflnwb, coaep1Kaw~eft 3alaaimpyo, KaKc HOpMaJlbHbWA aeAHTeA b.
n foc-ximee CB01ACTB0OHopeuienieT co6oii Tag Ha33ba3mp!o x ap a KT e pH C T H~qecicyta noArpynny.