U matematici p grupoyu de p proste chislo nazivayetsya grupa v yakij poryadok kozhnogo elementa ye stepenem chisla p tobto dlya kozhnogo elementa g isnuye naturalne chislo n sho gpn 1 i dlya vsih dodatnih m lt pn element gm ne dorivnyuye nejtralnomu Yaksho grupa skinchenna to yiyi poryadok todi tezh rivnij deyakomu stepenyu chisla p oskilki zgidno teorem Silova kozhna p pidgrupa zokrema i sama grupa maye mistitisya v deyakij pidgrupi Silova i tomu grupa sama ye svoyeyu pidgrupoyu Silova tobto yiyi poryadok ye stepenem chisla p V osnovnomu interes stanovlyat same skinchenni p grupi Zmist 1 Centr p grupi 1 1 Dovedennya 2 Vlastivosti 3 Skinchenni p grupi nevelikih poryadkiv 3 1 Chislo riznih UNIQ postMath 00000012 QINU grupp poryadku UNIQ postMath 00000013 QINU 3 2 p grupi poryadku pn asimptotika 4 Div takozh 5 DzherelaCentr p grupi RedaguvatiOdniyeyu z najvazhlivishih vlastivostej skinchennih p grup ye taka teorema Centr netrivialnoyi skinchennoyi p grupi ye netrivialnoyu grupoyu Dovedennya Redaguvati Vizmemo deyaku p grupu G G p k displaystyle G p k nbsp i zadamo diyu grupi G na mnozhini G ϕ G G G ϕ g x g x g 1 displaystyle phi colon G times G to G phi g x gxg 1 nbsp Spershu dovedemo sho orbita dovilnogo elementa skladayetsya lishe z togo elementa todi i lishe todi koli cej element nalezhit do centru grupi x G G x 1 x Z G displaystyle forall x in G G x 1 iff x in Z G nbsp Vizmemo dovilnij g G displaystyle g in G nbsp Todi g x g 1 x g g 1 x g x g 1 g g 1 x x Z G displaystyle gxg 1 x gg 1 x iff gxg 1 gg 1 x iff x in Z G nbsp Dali dovedemo sho yaksho deyaka orbita maye bilsh nizh odin element to yiyi poryadok dilitsya na p x G G x gt 1 p G x displaystyle forall x in G G x gt 1 Rightarrow p G x nbsp Pripustimo sho dlya x G displaystyle x in G nbsp mayemo G x gt 1 displaystyle G x gt 1 nbsp Oskilki stabilizator G x displaystyle G x nbsp ye pidgrupoyu G to zgidno z teoremoyu Lagranzha kilkist jogo elementiv dilit kilkist elementiv G otzhe G x p l l gt 0 displaystyle G x p l l gt 0 nbsp Dali G x G G x G G x p k p l p k l displaystyle G x G G x frac G G x frac p k p l p k l nbsp G ye ob yednannyam orbit G G x G x 1 G x G x gt 1 G x displaystyle G bigcup G x bigcup G x 1 G x cup bigcup G x gt 1 G x nbsp Zvidsi otrimuyemo p k G G x 1 G x G x gt 1 G x Z G i 1 s p a i displaystyle p k G sum limits G x 1 G x sum limits G x gt 1 G x Z G sum i 1 s p a i nbsp de s kilkist orbit sho mistyat bilshe odnogo elementa a vsi ai bilshi vid nulya Z ostannoyi formuli oderzhuyemo sho Z G dilitsya na p Vlastivosti RedaguvatiYaksho H displaystyle H nbsp normalna v P displaystyle P nbsp to H Z P gt 1 displaystyle H cap Z P gt 1 nbsp Cya vlastivist oderzhuyetsya z teoremi pro centr yaksho vrahuvati sho bud yaka pidgrupa p grupi sama ye p grupoyu i sho normalna pidgrupa invariantna do spryazhen Tomu v poperednomu dovedenni mozhna vzyati H zamist P i H Z P displaystyle H cap Z P nbsp zamist Z P Usi p grupi ye nilpotentnimi Skinchenni p grupi nevelikih poryadkiv RedaguvatiDiv takozh Skinchenna p grupa Chislo riznih p displaystyle p nbsp grupp poryadku p n displaystyle p n nbsp Redaguvati Chislo neizomorfnih grup poryadku p displaystyle p nbsp rivne 1 grupa C p displaystyle C p nbsp Chislo neizomorfnih grup poryadku p 2 displaystyle p 2 nbsp rivne 2 grupi C p 2 displaystyle C p 2 nbsp i C p C p displaystyle C p times C p nbsp Chislo neizomorfnih grup poryadku p 3 displaystyle p 3 nbsp rivne 5 z nih tri abelevi C p 3 displaystyle C p 3 nbsp C p 2 C p displaystyle C p 2 times C p nbsp C p C p C p displaystyle C p times C p times C p nbsp i dvi neabelevi pri p gt 2 displaystyle p gt 2 nbsp E p 3 displaystyle E p 3 nbsp i E p 3 displaystyle E p 3 nbsp pri p 2 D 8 displaystyle D 8 nbsp Q 8 displaystyle Q 8 nbsp Chislo neizomorfnih grup poryadku p 4 displaystyle p 4 nbsp rivne 15 pri p gt 2 displaystyle p gt 2 nbsp chislo grup poryadku 2 4 displaystyle 2 4 nbsp rivne 14 Chislo neizomorfnih grup poryadku p 5 displaystyle p 5 nbsp rivne 2 p 61 2 G C D p 1 3 G C D p 1 4 displaystyle 2p 61 2GCD p 1 3 GCD p 1 4 nbsp pri p 5 displaystyle p geq 5 nbsp Chislo grup poryadku 2 5 displaystyle 2 5 nbsp rivne 51 chislo grup poryadku 3 5 displaystyle 3 5 nbsp rivne 67 Chislo neizomorfnih grup poryadku p 6 displaystyle p 6 nbsp rivne 3 p 2 39 p 344 24 G C D p 1 3 11 G C D p 1 4 2 G C D p 1 5 displaystyle 3p 2 39p 344 24GCD p 1 3 11GCD p 1 4 2GCD p 1 5 nbsp pri p 5 displaystyle p geq 5 nbsp Chislo grup poryadku 2 6 displaystyle 2 6 nbsp rivne 267 chislo grup poryadku 3 6 displaystyle 3 6 nbsp rivne 504 Chislo neizomorfnih grup poryadku p 7 displaystyle p 7 nbsp rivne 3 p 5 12 p 4 44 p 3 170 p 2 707 p 2455 4 p 2 44 p 291 G C D p 1 3 p 2 19 p 135 G C D p 1 4 3 p 31 G C D p 1 5 4 G C D p 1 7 5 G C D p 1 8 G C D p 1 9 displaystyle 3p 5 12p 4 44p 3 170p 2 707p 2455 4p 2 44p 291 GCD p 1 3 p 2 19p 135 GCD p 1 4 3p 31 GCD p 1 5 4GCD p 1 7 5GCD p 1 8 GCD p 1 9 nbsp pri p gt 5 displaystyle p gt 5 nbsp Chislo grup poryadku 2 7 displaystyle 2 7 nbsp rivne 2328 chislo grup poryadk 3 7 displaystyle 3 7 nbsp rivne 9310 chislo grup poryadku 5 7 displaystyle 5 7 nbsp rivne 34297 p grupi poryadku pn asimptotika Redaguvati Pri n displaystyle n rightarrow infty nbsp chislo neizomorfnih grup poryadku p n displaystyle p n nbsp asimptotichno rivne p 2 27 O n 1 3 n 3 displaystyle p 2 27 O n 1 3 n 3 nbsp Div takozh RedaguvatiTeoremi SilovaDzherela RedaguvatiKurosh A G Teoriya grupp 3 e izd Moskva Nauka 1967 648 s ISBN 5 8114 0616 9 ros Holl M Teoriya grupp Izdatelstvo inostrannoj literatury M 1962 Gorenstein D Finite groups N Y Harper and Row 1968 Dzhozef Rotman en An Introduction to the Theory of Groups 4th Springer Graduate Texts in Mathematics 1994 532 s ISBN 978 0387942858 angl Otrimano z https uk wikipedia org w index php title P grupa amp oldid 37842689
