Символом ?(G) обозначим множество всех простых делителей порядка группы G. Граф простых чисел (G) конечной группы G это простой граф с множеством вершин ?(G), в котором две раз-личные вершины p и q соединены ребром тогда и только тогда, когда в G


Чтобы посмотреть этот PDF файл с форматированием и разметкой, скачайте файл и откройте на своем компьютере.
Ñèáèðñêèéìàòåìàòè÷åñêèéæóðíàë Èþëüàâãóñò,2014.Òîì55,4 ÓÄÊ512.542 ODÕÀÐÀÊÒÅÐÈÇÀÖÈßÂÑÅÕÊÎÍÅ×ÍÛÕ ÍÅÀÁÅËÅÂÛÕÏÐÎÑÒÛÕÃÐÓÏÏ ÏÎÐßÄÊÎÂ,ÏÐÎÑÒÛÅÄÅËÈÒÅËÈ ÊÎÒÎÐÛÕÍÅÏÐÅÂÎÑÕÎÄßÒ 13 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ Àííîòàöèÿ. Äëÿâñÿêîéêîíå÷íîéíåàáåëåâîéïðîñòîéãðóïïû M ïîðÿäêà,ïðî- ñòûåäåëèòåëèêîòîðîãîíåïðåâîñõîäÿò13,íàéäåíî÷èñëîêëàññîâèçîìîðôíûõ êîíå÷íûõãðóïïòåõæåïîðÿäêàèìîäåëèñòåïåíåé,÷òîè M .Ïîëó÷åííûéðåçóëü- òàòÿâëÿåòñÿàíàëîãîìðåçóëüòàòàÀ.Â.Âàñèëüåâà[1]îðàñïîçíàâàåìîñòèòàêèõ ïðîñòûõãðóïïïîñïåêòðó.Âñèëóèçâåñòíîãîðåçóëüòàòàìûäîëæíûðàññìîòðåòü òîëüêîãðóïïû L 6 (3), U 4 (5), G 2 (4), L 5 (3), S 4 (8), U 6 (2)è O + 8 (3). Êëþ÷åâûåñëîâà: êîíå÷íàÿïðîñòàÿãðóïïà,OD-õàðàêòåðèçàöèÿ,ãðóïïàëèåâà òèïà. 1.Ââåäåíèå Ïóñòü G êîíå÷íàÿãðóïïà.Ñèìâîëîì  ( G )îáîçíà÷èììíîæåñòâîâñåõ ïðîñòûõäåëèòåëåéïîðÿäêàãðóïïû G .Ãðàôïðîñòûõ÷èñåë € ( G )êîíå÷íîé ãðóïïû G ýòîïðîñòîéãðàôñìíîæåñòâîìâåðøèí  ( G ),âêîòîðîìäâåðàç- ëè÷íûåâåðøèíû p è q ñîåäèíåíûðåáðîìòîãäàèòîëüêîòîãäà,êîãäàâ G èìå- åòñÿýëåìåíòïîðÿäêà pq .Âäàííîéñòàòüåðàññìîòðåíûêîíå÷íûåíåàáåëåâû ïðîñòûåãðóïïû G ñîñâîéñòâîì  ( G ) f 2 ; 3 ; 5 ; 7 ; 11 ; 13 g .Ìíîæåñòâîâñåõòà- êèõãðóïïîáîçíà÷àåòñÿñèìâîëîì S 13 .Èñïîëüçóÿêëàññèôèêàöèþêîíå÷íûõ ïðîñòûõãðóïï,íåòðóäíîïîëó÷èòüïîëíûéñïèñîêèç S 13 .Ñîãëàñíî[2]èìååòñÿ 55òàêèõãðóïï,êîòîðûåïåðå÷èñëåíûâòàáë.1. Îïðåäåëåíèå1.1. Ïóñòü G êîíå÷íàÿãðóïïàè  ( G )= f p 1 ;p 2 ;:::;p k g . Äëÿ p 2  ( G )ïóñòüdeg( p )= jf q 2  ( G ) j p  q gj åñòüñòåïåíüâåðøèíû p âãðàôå € ( G ).Ïîëîæèì D ( G )=(deg( p 1 ) ; deg( p 2 ) ;:::; deg( p k )),ãäå p 1 p 2  p k ; D ( G )íàçûâàåòñÿ ìîäåëüþñòåïåíåéãðóïïû G . Äëÿêîíå÷íîéãðóïïû G îáîçíà÷èìñèìâîëîì h OD ( G )÷èñëîêëàññîâèçî- ìîðôíûõêîíå÷íûõãðóïï S òàêèõ,÷òî j G j = j S j è D ( G )= D ( S ).Âòåðìèíàõ ôóíêöèè h OD ãðóïïû G êëàññèôèöèðóþòñÿñëåäóþùèìîáðàçîì. Îïðåäåëåíèå1.2. Ãðóïïà G íàçûâàåòñÿ k - êðàòíî OD- õàðàêòåðèçóåìîé , åñëèñóùåñòâóþòâòî÷íîñòè k íåèçîìîðôíûõãðóïï S òàêèõ,÷òî j G j = j S j è D ( G )= D ( S ).ÎäíîêðàòíîOD-õàðàêòåðèçóåìàÿãðóïïàíàçûâàåòñÿïðîñòî OD- õàðàêòåðèçóåìîé. Îñíîâíîéöåëüþäàííîéñòàòüèÿâëÿåòñÿíàõîæäåíèå h OD ( G )äëÿâñåõýëå- ìåíòîâ S 13 .Íîâñèëóòàáë.2äîñòàòî÷íîðàññìîòðåòüòîëüêîãðóïïû L 6 (3), U 4 (5), G 2 (4), L 5 (3), S 4 (8), U 6 (2)è O + 8 (3). c 2014ÍîñðàòïóðÏ.,ÄàðàôøåõÌ.Ð. OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 807 Òàáëèöà1. Íåàáåëåâûïðîñòûåãðóïïûïîðÿäêîâ, ïðîñòûåäåëèòåëèêîòîðûõíåïðåâîñõîäÿò13 P j P j j Out( P ) j A 5 2 2  3  5 2 A 6 2 3  3 2  5 4 U 4 (2) 2 6  3 4  5 2 L 2 (7) 2 3  3  7 2 L 2 (8) 2 3  3 2  7 3 U 3 (3) 2 5  3 3  7 2 A 7 2 3  3 2  5  7 2 L 2 (49) 2 4  3  5 2  7 2 4 U 3 (5) 2 4  3 2  5 3  7 6 L 3 (4) 2 6  3 2  5  7 12 A 8 2 6  3 2  5  7 2 A 9 2 6  3 4  5  7 2 J 2 2 7  3 3  5 2  7 2 A 10 2 7  3 4  5 2  7 2 U 4 (3) 2 7  3 6  5  7 8 S 4 (7) 2 8  3 2  5 2  7 4 2 P j P j j Out( P ) j S 6 (2) 2 9  3 4  5  7 1 O + 8 (2) 2 12  3 5  5 2  7 6 L 2 (11) 2 2  3  5  11 2 M 11 2 4  3 2  5  11 1 M 12 2 6  3 3  5  11 2 U 5 (2) 2 10  3 5  5  11 2 M 22 2 7  3 2  5  7  11 2 A 11 2 7  3 4  5 2  7  11 2 M c L 2 6  3 6  5 3  7  11 2 HS 2 9  3 2  5 3  7  11 2 A 12 2 9  3 5  5 2  7  11 2 U 6 (2) 2 15  3 6  5  7  11 6 L 3 (3) 2 4  3 3  13 2 L 2 (25) 2 3  3  5 2  13 4 U 3 (4) 2 6  3  5 2  13 4 S 4 (5) 2 6  3 2  5 4  13 2 P j P j j Out( P ) j L 4 (3) 2 7  3 6  5  13 4 2 F 4 (2) 0 2 11  3 3  5 2  13 2 L 2 (13) 2 2  3  7  13 2 L 2 (27) 2 2  3 3  7  13 6 L 2 (64) 2 6  3 2  5  7  13 6 Sz (8) 2 6  5  7  13 3 G 2 (3) 2 6  3 6  7  13 2 L 3 (9) 2 7  3 6  5  7  13 4 3 D 4 (2) 2 12  3 4  7 2  13 3 G 2 (4) 2 12  3 3  5 2  7  13 2 S 4 (8) 2 12  3 4  5  7 2  13 6 A 13 2 9  3 5  5 2  7  11  13 2 P j P j j Out( P ) j S 6 (3) 2 9  3 9  5  7  13 2 O 7 (3) 2 9  3 9  5  7  13 2 U 4 (5) 2 7  3 4  5 6  7  13 4 A 14 2 10  3 5  5 2  7 2  11  13 2 L 5 (3) 2 9  3 10  5  11 2  13 2 Suz 2 13  3 7  5 2  7  11  13 2 A 15 2 10  3 6  5 3  7 2  11  13 2 O + 8 (3) 2 12  3 12  5 2  7  13 24 A 16 2 14  3 6  5 3  7 2  11  13 2 Fi 22 2 17  3 9  5 2  7  11  13 2 L 6 (3) 2 11  3 15  5  7  11 2  13 2 4 2.Ïðåäâàðèòåëüíûåñâåäåíèÿ Äëÿïðîèçâîëüíîéãðóïïû G ïóñòü ! ( G )ìíîæåñòâîïîðÿäêîâýëåìåíòîâ ãðóïïû G ,ãäåêàæäûéâîçìîæíûéïîðÿäîêâñòðå÷àåòñÿâ ! ( G )îäèíðàçíåçàâè- ñèìîîòòîãî,ñêîëüêîýëåìåíòîâýòîãîïîðÿäêàèìååòñÿâ G .Ýòîìíîæåñòâîçà- ìêíóòîè÷àñòè÷íîóïîðÿäî÷åíîäåëèìîñòüþ,ïîñêîëüêóîíîîäíîçíà÷íîîïðåäå- ëÿåòñÿñâîèìèìàêñèìàëüíûìèýëåìåíòàìè.Ìíîæåñòâîìàêñèìàëüíûõýëåìåí- òîââ ! ( G )îáîçíà÷àåòñÿñèìâîëîì  ( G ).×èñëîñâÿçíûõêîìïîíåíòâ € ( G )îáî- çíà÷àåòñÿ÷åðåç t ( G ).Ïóñòü  i =  i ( G ),1  i  t ( G ), i -åñâÿçíûåêîìïîíåíòû ãðàôà € ( G ).Äëÿãðóïïû÷åòíîãîïîðÿäêàïîëàãàåì2 2  1 ( G ).Ñèìâîëîì  ( n ) îáîçíà÷èììíîæåñòâîâñåõïðîñòûõäåëèòåëåéíàòóðàëüíîãî÷èñëà n .Òîãäà j G j 808 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ ìîæíîïðåäñòàâèòüââèäåïðîèçâåäåíèÿ m 1 ;m 2 ;:::;m t ( G ) ,ãäå m i ïîëîæè- òåëüíûåöåëûå÷èñëàñîñâîéñòâîì  ( m i )=  i .Ýòè÷èñëàíàçûâàþòñÿ ïîðÿä- êîâûìèêîìïîíåíòàìèãðóïïû G .Áóäåìïèñàòü OC ( G )= f m 1 ;m 2 ;:::;m t ( G ) g èíàçûâàòü OC ( G ) ìíîæåñòâîìïîðÿäêîâûõêîìïîíåíòãðóïïû G .Ìíîæå- ñòâîêîìïîíåíòãðàôàïðîñòûõ÷èñåëãðóïïû G îáîçíà÷àåòñÿñèìâîëîì T ( G )= f  i ( G ) j i =1 ; 2 ;:::;t ( G ) g . Âòàáë.2âûïèñàíûêîíå÷íûåïðîñòûåãðóïïû,ïðîêîòîðûåâíàñòîÿùåå âðåìÿèçâåñòíî,÷òîîíèOD-õàðàêòåðèçóåìûèëèäâóêðàòíîOD-õàðàêòåðèçóåìû. Òàáëèöà2. k -ÊðàòíîOD-õàðàêòåðèçóåìûåãðóïïû, k =1 ; 2 M Óñëîâèÿíà M h OD ( M ) Ññûëêè A n n = p;p +1 ;p +2( p ïðîñòîå) 1 [3,4] n = p +3, p 2  (100!) �f 7 g 1 [58] n =10 2 [9] L 2 ( q ) q 6 =2 ; 3 1 [3,4,10,11] L 3 ( q )  � q 2 + q +1 d  =1, d =(3 ;q � 1) 1 [3] U 3 ( q )  � q 2 � q +1 d  =1, d =(3 ;q +1), q� 5 1 [3] L 4 ( q ) q =5 ; 7 1 [12] L 3 (9) 1 [13] U 3 (5) 1 [14] U 4 (7) 1 [12] L n (2) n = p èëè p +1,ãäå2 p � 1ïðîñòîå 1 [12] R ( q ) j  ( q  p 3 q +1) j =1, q =3 2 m +1 , m  1 1 [3] Sz ( q ) q =2 2 n +1  8 1 [3,4] B 3 (3)  = O 7 (3) 2 [3] C 3 (3)  = S 6 (3) 2 [3] M Ñïîðàäè÷åñêàÿïðîñòàÿãðóïïà 1 [3] M j  ( M ) j =4, M 6 = A 10 1 [15] M j M j 10 8 , M 6 = A 10 , U 4 (2) 1 [16] C r (3), B r (3) r íå÷åòíîåïðîñòîå,  � 3 r � 1 2  =1 2 [17] C n ( q ), B n ( q ) n =2 m  4, q íå÷åòíîå,  � q n +1 2  =1 2 [17] C 2 ( q )  = B 2 ( q ) q íå÷åòíîå, q 6 =3, j  ( q 2 +1) j =1 1 [17] C n ( q ), B n ( q ) n =2 m  2, q ÷åòíîå,( n;q ) 6 =(2 ; 2), j  ( q n +1) j =1 1 [17] 3.Ýëåìåíòàðíûåðåçóëüòàòû Ëåììà3.1 [18]. Ïóñòü G êîíå÷íàÿãðóïïàè j  ( G ) j 3 .Åñëèñóùå- ñòâóþòïðîñòûå÷èñëà r;s;t 2  ( G ) òàêèå,÷òî f tr;ts;rs g\ ! ( G )= ? ,òî G íåðàçðåøèìà. Îïðåäåëåíèå3.1. Ãðóïïà G íàçûâàåòñÿ2- ôðîáåíèóñîâîé ,åñëèñóùåñòâó- åòíîðìàëüíûéðÿä1 /H/K/G òàêîé,÷òî K è G H ôðîáåíèóñîâûãðóïïûñ ÿäðàìè H è K H ñîîòâåòñòâåííî. OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 809 Ëåììà3.2 [19]. Ïóñòü G  2 -ôðîáåíèóñîâàãðóïïà÷åòíîãîïîðÿäêà,èìå- þùàÿíîðìàëüíûéðÿä 1 /H/K/G òàêîé,÷òî K è G H ôðîáåíèóñîâûãðóïïû ñÿäðàìè H è K H ñîîòâåòñòâåííî.Òîãäà (a) t ( G )=2 è T ( G )=   1 ( G )=  ( H ) [  � G K  ; 2 ( G )=  � K H  ; (b) G K è K H öèêëè÷åñêèåãðóïïû, G K j Aut � K H  è � G K ; K H  =1 ; (c) H íèëüïîòåíòíàÿãðóïïàè G ðàçðåøèìàÿãðóïïà. Ñëåäóþùèåëåììûïîëåçíûïðèðàáîòåñôðîáåíèóñîâûìèãðóïïàìè. Ëåììà3.3 [20,21]. Ïóñòü G ôðîáåíèóñîâàãðóïïàñäîïîëíåíèåì H è ÿäðîì K .Ñïðàâåäëèâûñëåäóþùèåóòâåðæäåíèÿ. (a) K íèëüïîòåíòíàÿãðóïïà. (b) j K j 1(mod j H j ) . (c) Âñÿêàÿïîäãðóïïàãðóïïû H ïîðÿäêà pq ,ãäå p , q  ( íåîáÿçàòåëü- íîðàçëè÷íûå ) ïðîñòûå÷èñëà,öèêëè÷åñêàÿ.Â÷àñòíîñòè,âñÿêàÿñèëîâñêàÿ ïîäãðóïïàíå÷åòíîãîïîðÿäêàâ H öèêëè÷åñêàÿ,èñèëîâñêàÿ 2 -ïîäãðóïïàâ H ëèáîöèêëè÷åñêàÿ,ëèáîÿâëÿåòñÿîáîáùåííîéãðóïïîéêâàòåðíèîíîâ.Åñ- ëè H íåðàçðåøèìàÿãðóïïà,òîâ H èìååòñÿïîäãðóïïàèíäåêñàíåáîëüøå 2 ,èçîìîðôíàÿ Z  SL (2 ; 5) ,ãäå Z èìååòöèêëè÷åñêèåñèëîâñêèåïîäãðóïïûè  ( Z ) \f 2 ; 3 ; 5 g = ? .Â÷àñòíîñòè, 15 ; 20 = 2 ! ( H ) .Åñëè H ðàçðåøèìàè O ( H )=1 , òîëèáî H  2 -ãðóïïà,ëèáî H èìååòïîäãðóïïóèíäåêñàíåáîëüøå 2 ,èçîìîðô- íóþ SL (2 ; 3) . Ëåììà3.4 [19]. Ïóñòü G ôðîáåíèóñîâàãðóïïà÷åòíîãîïîðÿäêà,ãäå H è K ôðîáåíèóñîâîäîïîëíåíèåèôðîáåíèóñîâîÿäðîãðóïïû G ñîîòâåòñòâåííî. Òîãäà t ( G )=2 è T ( G )= f  ( H ) ; ( K ) g . Ñòðîåíèåêîíå÷íîéãðóïïûñíåñâÿçíûìãðàôîìïðîñòûõ÷èñåëîïèñûâàåòñÿ ñëåäóþùåéëåììîé. Ëåììà3.5 [22,23]. Ïóñòü G êîíå÷íàÿãðóïïàè t ( G )  2 .Òîãäà G ÿâëÿåòñÿîäíîéèçñëåäóþùèõãðóïï: (a) G ôðîáåíèóñîâàèëè 2 -ôðîáåíèóñîâàãðóïïà; (b) G èìååòíîðìàëüíûéðÿä 1 E H/K E G òàêîé,÷òî H è G K   1 - ãðóïïûè K H íåàáåëåâàïðîñòàÿãðóïïà,ãäå  1 êîìïîíåíòàãðàôàïðîñòûõ ÷èñåë,ñîäåðæàùàÿ 2 , H íèëüïîòåíòíàÿãðóïïàè G H j Aut � K H  .Êðîìåòîãî, âñÿêàÿêîìïîíåíòàíå÷åòíîãîïîðÿäêà G òàêæåÿâëÿåòñÿêîìïîíåíòîéíå÷åòíîãî ïîðÿäêàâ K H . Ñëåäóþùàÿëåììàçàèìñòâîâàíàèç[24]. Ëåììà3.6. Ïóñòü S = P 1  P 2  P r ,ãäå P i èçîìîðôíûåíåàáåëåâû ïðîñòûåãðóïïû.Òîãäà Aut( S )  = (Aut( P 1 )  Aut( P 2 )  Aut( P r ))  S r . Èñïîëüçóÿðåçóëüòàòû,ñîáðàííûåâòàáë.1,ïîëó÷àåìñëåäóþùóþëåììó. Ëåììà3.7. Åñëè S 2 S 13 è S  Sz (8) ,òî f 2 ; 3 g  ( S ) ,àåñëè Out( S ) 6 =1 , òî  (Out( S )) f 2 ; 3 g . 4.Îñíîâíûåðåçóëüòàòû ÂýòîìðàçäåëåìûðàññìîòðèìOD-õàðàêòåðèçóåìîñòüïðîñòûõãðóïï L 6 (3), U 4 (5), G 2 (4), L 5 (3), S 4 (8), U 6 (2)è O + 8 (3). Ñîãëàñíî[25]èìååì  ( L 6 (3))= f 36 ; 78 ; 80 ; 104 ; 120 ; 121 ; 182 g ,îòêóäàâûâî- äèì,÷òî D ( L 6 (3))=(4 ; 3 ; 2 ; 2 ; 0 ; 3). 810 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ Òåîðåìà4.1. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( L 6 (3))= (4 ; 3 ; 2 ; 2 , 0 ; 3) è j G j = j L 6 (3) j .Òîãäà G  = L 6 (3) . Äîêàçàòåëüñòâî. Èìååì j L 6 (3) j =2 11  3 15  5  7  11 2  13 2 .Ó÷èòûâàÿ, ÷òî D ( G )=(4 ; 3 ; 2 ; 2 ; 0 ; 3),ðàññìîòðèìðàçëè÷íûåâîçìîæíîñòèäëÿìíîæåñòâà ðåáåðãðàôà € ( G )èïîëó÷èìñëåäóþùèåñëó÷àèäëÿ € ( G ): f 2  3 ; 2  5 ; 2  7 ; 2  13 ; 3  5 ; 3  13 ; 7  13;11 g èëè f 2  3 ; 2  5 ; 2  7 ; 2  13 ; 3  7 ; 3  13 ; 5  13;11 g . Ñëó÷àé(1) .Åñëè € ( G )= f 2  3 ; 2  5 ; 2  7 ; 2  13 ; 3  5 ; 3  13 ; 7  13;11 g ,òî f 11  7 ; 11  5 ; 7  5 g\ ! ( G )= ? ,Ïîýòîìóâñèëóëåììû3.1 G íåðàçðåøèìà, îòêóäàñëåäóåò,÷òî G íåÿâëÿåòñÿ2-ôðîáåíèóñîâîéãðóïïîéïîëåììå3.2(c). Ïðåäïîëîæèì,÷òî G íåðàçðåøèìàÿôðîáåíèóñîâàãðóïïàñôðîáåíèó- ñîâûìäîïîëíåíèåì H èôðîáåíèóñîâûìÿäðîì K ñîîòâåòñòâåííî.Èñïîëüçóÿ îáîçíà÷åíèÿëåììû3.3(c),ïîëó÷àåì13 2  ( Z ),îòêóäàñëåäóåò,÷òîâ H 0 èìååòñÿ ýëåìåíòïîðÿäêà13  5;ïðîòèâîðå÷èå.Çíà÷èò, G íåÿâëÿåòñÿíèôðîáåíèóñîâîé, íè2-ôðîáåíèóñîâîéãðóïïîé. Âñèëóëåììû3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 -ãðóïïà.Ïîýòîìó K H  G H  Aut � K H  .Òàêêàê11 = 2  1 ( G ),èìååì11 2  � K H  . Ñëåäîâàòåëüíî, K H 2 S 13 è11 j K H .Âñèëóòàáë.1ãðóïïà K H èçîìîðôíà L 6 (3), L 5 (3), U 5 (2), L 2 (11), M 11 , M 12 èëè M 22 . Åñëè K H  = M 11 ,òî M 11  G H  Aut( M 11 ),òàêêàê G H  Aut � K H  .Ñëåäîâà- òåëüíî, j H j =2 7  3 13  7  11  13 2 .Ïîñêîëüêó H íèëüïîòåíòíà,13  11â € ( G ); ïðîòèâîðå÷èå. Àíàëîãè÷íîìîæíîïîêàçàòü,÷òî K H  U 5 (2) ;L 2 (11) ;M 12 ;M 22 . Åñëè K H  = L 5 (3),òî L 5 (3)  G H  Aut( L 5 (3),òàêêàê G H  Aut � K H  .Ñëå- äîâàòåëüíî, j H j =2 2  3 5  7  13èëè j H j =2  3 5  7  13.Ïóñòü H 7 2 Syl 7 ( H ) è G 11 2 Syl 11 ( G ).Òîãäà H 7 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäà H 7 E G .Òàêèìîáðàçîì, A = H 7  G 11 ÿâëÿåòñÿïîäãðóïïîéâ G .Èìååì H 7 E A , èïîòåîðåìåÑèëîâà G 11 E A ,îòêóäà A = H 7  G 11 ,÷òîäàåò7  11;ïðîòèâîðå- ÷èå.Ïîýòîìó K H  = L 6 (3)èòîãäàèç j G j = j L 6 (3) j ñëåäóåò,÷òî j H j =1.Çíà÷èò, G  = L 6 (3). Ñëó÷àé(2) . € ( G )= f 2  3 ; 2  5 ; 2  7 ; 2  13 ; 3  7 ; 3  13 ; 5  13;11 g , òîãäà f 11  7 ; 11  5 ; 7  5 g\ ! ( G )= ? .Ïîýòîìóâñèëóëåììû3.1 G íåðàçðåøèìà, îòêóäàâûòåêàåò,÷òî G íåÿâëÿåòñÿ2-ôðîáåíèóñîâîéãðóïïîéïîëåììå3.2(c). Ïóñòü G íåðàçðåøèìàÿôðîáåíèóñîâàãðóïïàñôðîáåíèóñîâûìäîïîë- íåíèåì H èôðîáåíèóñîâûìÿäðîì K .Èñïîëüçóÿòåæåîáîçíà÷åíèÿ,÷òîâ ëåììå3.3(c),ïîëó÷àåì13 2  ( Z ),èïîòîìóâ H 0 èìååòñÿýëåìåíòïîðÿäêà13  7; ïðîòèâîðå÷èå.Çíà÷èò, G íåÿâëÿåòñÿíèôðîáåíèóñîâîé,íè2-ôðîáåíèóñîâîé ãðóïïîé. Âñèëóëåììû3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 -ãðóïïà.Ñëåäîâàòåëüíî, K H  G H  Aut � K H  .Òàêêàê11 = 2  1 ( G ),èìååì 11 2  � K H  .Ïîýòîìó K H 2 S 13 è11 j K H .Èçòàáë.1ïîëó÷àåì,÷òî K H èçîìîðôíà L 6 (3), L 5 (3), U 5 (2), L 2 (11), M 11 , M 12 èëè M 22 .Åñëè K H  = M 11 ,òî M 11  G H  Aut( M 11 ),òàêêàê G H  Aut � K H  .Ñëåäîâàòåëüíî, j H j =2 7  3 13  7  11  13 2 . ñèëóíèëüïîòåíòíîñòè H èìååì13  11â € ( G );ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  U 5 (2) ;L 2 (11) ;M 12 ;M 22 . Åñëè K H  = L 5 (3),òî L 5 (3)  G H  Aut( L 5 (3)),òàêêàê G H  Aut � K H  .Ñëåäî- OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 811 âàòåëüíî, j H j =2 2  3 5  7  13èëè j H j =2  3 5  7  13.Âñèëóíèëüïîòåíòíîñòè H 13  7â € ( G );ïðîòèâîðå÷èå. Ïîýòîìó K H  = L 6 (3),òîãäàèç j G j = j L 6 (3) j ñëåäóåò,÷òî j H j =1.Çíà÷èò, G  = L 6 (3).Ïîëó÷àåìïðîòèâîðå÷èå,òàêêàê € ( G ) 6 = € ( L 6 (3)).Òåìñàìûìâòîðàÿ âîçìîæíîñòüïðèâîäèòêïðîòèâîðå÷èþ,îòêóäàñëåäóåò,÷òîãðóïïà L 6 (3)OD- õàðàêòåðèçóåìà.  Ñîãëàñíî[1]èìååì  ( U 4 (5))= f 63 ; 60 ; 52 ; 24 g ,òàê÷òî D ( U 4 (5))=(3 ; 3 ; 2 ; 1 ; 1). Òåîðåìà4.2. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( U 4 (5))= (3 ; 3 ; 2 ; 1 ; 1) è j G j = j U 4 (5) j .Òîãäà G  = U 4 (5) . Äîêàçàòåëüñòâî. Èìååì j U 4 (5) j =2 7  3 4  5 6  7  13è D ( U 4 (5))=(3 ; 3 ; 2 ; 1 ; 1). Òàêêàê D ( G )= D ( U 4 (5))=(3 ; 3 ; 2 ; 1 ; 1)è € ( U 4 (5))ñâÿçíûéãðàô,ëåãêî âèäåòü,÷òî € ( G )= € ( U 4 (5))è f 2 ; 3 ; 5 ; 7 ; 13 ; 6 ; 10 ; 26 ; 15 ; 21 g ! ( G ).Ïîýòîìó f 5 ; 7 ; 13 g íåçàâèñèìîåìíîæåñòâîâ € ( G )è f 2 ; 7 g 2-íåçàâèñèìîåìíîæåñòâî. Ðàçîáüåìäîêàçàòåëüñòâîíàñåðèþëåìì. Ëåììà4.1. Ïóñòü K ìàêñèìàëüíàÿíîðìàëüíàÿðàçðåøèìàÿïîäãðóïïà ãðóïïû G .Òîãäà K  f 2 ; 3 g -ãðóïïà.Â÷àñòíîñòè, G íåðàçðåøèìà. Äîêàçàòåëüñòâî. Ïóñòü p ïðîñòîéäåëèòåëüïîðÿäêàãðóïïû K è K p  ñèëîâñêàÿ p -ïîäãðóïïàâ K .Òàêêàê K E G ,âñèëóàðãóìåíòàÔðàòòèíèèìååì G = KN G ( K p ).Äàëååðàññìîòðèìíåñêîëüêîñëó÷àåâ. Ñëó÷àé(1) : p =13.Âýòîìñëó÷àå K 13 öèêëè÷åñêàÿãðóïïïîðÿäêà13, ïîýòîìó N = N G ( K 13 ) C G ( K 13 )  Aut( K 13 )  = Z 12 .Ñëåäîâàòåëüíî, j N j äåëèòåëü÷èñ- ëà12.Íîdeg(13)=1â € ( G )è13ñîåäèíÿåòñÿòîëüêîñ2,àçíà÷èò, C G ( K 13 ) f 2 ; 13 g -ãðóïïà.Òåìñàìûì N G ( K 13 ) f 2 ; 3 ; 13 g -ãðóïïà;òîãäà f 5 ; 7 ; 13 g  ( K ) èèç G = KN G ( K 13 )ïîëó÷àåì,÷òî7 jj K j .Òàêêàêãðóïïà K ïðåäïîëàãàåòñÿ ðàçðåøèìîé,ìîæíîðàññìîòðåòü f 7 ; 13 g -õîëëîâóïîäãðóïïóâ K ,êîòîðàÿèìååò ïîðÿäîê7  13èäîëæíàáûòüöèêëè÷åñêîé.Èçýòîãîâûòåêàåò,÷òî7  13; ïðîòèâîðå÷èå. Ñëó÷àé (2): p 2f 5 ; 7 g , K p 2 Syl p ( K ).ÑíîâàèñïîëüçóÿàðãóìåíòÔðàò- òèíè,ïîëó÷àåì,÷òî G = KN G ( K p )è13 jj N G ( K p ) j .Ïóñòü L ïîäãðóïïà ïîðÿäêà13ãðóïïû N G ( K p ).Ïîñêîëüêó L íîðìàëèçóåò K p ,ãðóïïà G ñîäåðæèò ïîäãðóïïóïîðÿäêà13  p èýòîïðèâîäèòêïðîòèâîðå÷èþòàêæå,êàêèâûøå, ïîòîìó÷òî p - (13 � 1). Ñëó÷àé (3): K  f 2 ; 3 g -ãðóïïà.Êðîìåòîãî,òàêêàê K 6 = G ,òî G íåðàç- ðåøèìà.Ýòîçàâåðøàåòäîêàçàòåëüñòâî.  Ëåììà4.2. Ôàêòîð-ãðóïïà G K ÿâëÿåòñÿïî÷òèïðîñòîéãðóïïîé.Áîëåå òîãî, S  G K  Aut( S ) ,ãäå S  = U 4 (5) . Äîêàçàòåëüñòâî. Ïóñòü H := G K , S :=Soc( H ),ãäåSoc( H )îçíà÷àåòöî- êîëüãðóïïû H ,ò.å.ïîäãðóïïóâ H ,ïîðîæäåííóþìíîæåñòâîìâñåõìèíèìàëü- íûõíîðìàëüíûõïîäãðóïïâ H .Òîãäà S  = P 1  P 2  P r ,ãäåâñå P i  íåàáåëåâûïðîñòûåãðóïïûè S  H  Aut( S ).Íèæåïîêàæåì,÷òî r =1è P 1  = U 4 (5). Ïðåäïîëîæèì,÷òî r  2.Âýòîìñëó÷àåëåãêîâèäåòü,÷òî13 - j S j ,ïîñêîëü- êóèíà÷åïîðÿäîêòîëüêîîäíîéãðóïïû P i äåëèòñÿíà13.Ïðåäïîëîæèì,÷òî 13 jj P 1 j .Òàêêàêòîëüêî13  2âûïîëíÿåòñÿâãðàôåïðîñòûõ÷èñåëãðóïïû G , òîãðóïïà P i , i  2,äîëæíàáûòü2-ãðóïïîé,÷òîïðîòèâîðå÷èòïðîñòîòåãðóï- ïû P i .Çíà÷èò,13 - j S j ,èäëÿêàæäîãî i èìååì P i 2 S 7 .Ñäðóãîéñòîðîíû, 812 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ èñïîëüçóÿëåììó4.1,çàìåòèì,÷òî13 2  ( H )   (Aut( S )).Òàêèìîáðàçîì, ìîæíîïðåäïîëàãàòü,÷òî13 jj Out( S ) j ,ïîñêîëüêó j Aut( S ) j = j S jj Out( S ) j .Íî Out( S )=Out( S 1 )  Out( S 2 )  Out( S k ),ãäåãðóïïû S j ÿâëÿþòñÿïðÿìû- ìèïðîèçâåäåíèÿìèèçîìîðôíûõãðóïï P i òàêèìè,÷òî S  = S 1  S 2  S k . Ïîýòîìóäëÿíåêîòîðîãî j ÷èñëî13äåëèòïîðÿäîêãðóïïûâíåøíèõàâòîìîð- ôèçìîâïðÿìîãîïðîèçâåäåíèÿ S j t èçîìîðôíûõïðîñòûõãðóïï P i .Òàêêàê P i 2 S 7 ,òî j Out( P i ) j íåäåëèòñÿíà13(ñìòàáë.1).Ïîëåììå3.6ïîëó÷à- åì,÷òî j Aut( S j ) j = j Aut( P i ) j t  t !.Ñëåäîâàòåëüíî, t  13,èïîòîìó÷èñëî (13!) 2  2 13 =2 23 äîëæíîäåëèòüïîðÿäîêãðóïïû G ;ïðîòèâîðå÷èå.Ïîýòîìó r =1è S = P 1 . Ââèäóëåìì3.7è4.1î÷åâèäíî,÷òî j S j =2 a  3 b  5 6  7  13,ãäå2  a  7è 1  b  4.Èñïîëüçóÿðåçóëüòàòû,ñîáðàííûåâòàáë.1,ïîëó÷àåì,÷òî S  = U 4 (5), èëåììàäîêàçàíà.  Ëåììà4.3. Ãðóïïà G èçîìîðôíàãðóïïå U 4 (5) . Äîêàçàòåëüñòâî. Ïîëåììå4.2 U 4 (5)  G K  Aut( U 4 (5)),îòêóäà G K  = U 4 (5)èëè U 4 (5):2,èëè U 4 (5):2 2 ,ïîñêîëüêó j Out( U 4 (5)) j =4.Âñëó÷àå G K  = U 4 (5),ðàññìàòðèâàÿïîðÿäêè,âûâîäèì,÷òî j K j =1è G  = U 4 (5),òàêêàê j G j = j U 4 (5) j .Âñëó÷àå G K  = U 4 (5):2,ðàññìàòðèâàÿïîðÿäêè,ïîëó÷àåì2 j K j = 1,÷òîíåâîçìîæíî.Âïîñëåäíåìñëó÷àå4 j K j =1,÷òîòàêæåíåâîçìîæíî.  Ñîãëàñíî[26]èìååì  ( G 2 (4))= f 8 ; 10 ; 12 ; 13 ; 15 ; 21 g ,îòêóäàâûâîäèì,÷òî D ( G 2 (4))=(2 ; 3 ; 2 ; 1 ; 0). Òåîðåìà4.3. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( G 2 (4))= (2 ; 3 ; 2 ; 1 ; 0) è j G j = j G 2 (4) j .Òîãäà G  = G 2 (4) . Äîêàçàòåëüñòâî. Èìååì j G 2 (4) j =2 12  3 3  5 2  7  13è D ( G 2 (4))=(2 ; 3 ; 2 ; 1 ; 0). Òàêæåèìååì € ( G )= f 2  3 ; 2  5 ; 3  5 ; 3  7;13 g ;òîãäà f 13  5 ; 13  7 ; 5  7 g\ ! ( G )= ? èïîëåììå3.1 G íåðàçðåøèìà,îòêóäàñëåäóåò,÷òî G íåÿâëÿåòñÿ 2-ôðîáåíèóñîâîéãðóïïîéâñèëóëåììû3.2(c). Ïðåäïîëîæèì,÷òî G íåðàçðåøèìàÿôðîáåíèóñîâàãðóïïàñôðîáåíèóñî- âûìäîïîëíåíèåì H èôðîáåíèóñîâûìÿäðîì K .Èñïîëüçóÿòåæåîáîçíà÷åíèÿ, ÷òîâëåììå3.3(c),ïîëó÷àåì,÷òî7 2  ( Z ).Ñëåäîâàòåëüíî,â H 0 èìååòñÿýëå- ìåíòïîðÿäêà7  5;ïðîòèâîðå÷èå.Çíà÷èò, G íåÿâëÿåòñÿíèôðîáåíèóñîâîé,íè 2-ôðîáåíèóñîâîéãðóïïîé. Ïîëåììå3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 -ãðóïïà.Ñëåäîâàòåëüíî, K H  G H  Aut � K H  .Òàêêàê13 = 2  1 ( G ),èìååì 13 2  � K H  .Ïîýòîìó K H 2 S 13 è13 j K H .Âñèëóòàáë.1èòîãîôàêòà,÷òî 11 - j G j ,ïîëó÷àåì,÷òî K H èçîìîðôíà L 2 (13), L 2 (25), L 2 (27), Sz (8), U 3 (4), L 2 (64), 2 F 4 (2) 0 , L 3 (3)èëè G 2 (4). Åñëè K H  = L 2 (13),òî L 2 (13)  G H  Aut( L 2 (13)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 10  3 2  5 2 èëè j H j =2 9  3 2  5 2 .Ïóñòü H 5 2 Syl 5 ( H ) è G 13 2 Syl 13 ( G ).Òîãäà H 5 char H E G âñèëóíèëüïîòåíòíîñòèãðóïïû H , îòêóäàñëåäóåò,÷òî H 5 E G .Òàêèìîáðàçîì, A = H 5  G 13 åñòüïîäãðóïïàâ G . Èìååì H 5 E A ,èïîòåîðåìåÑèëîâà G 13 E A .Çíà÷èò, A = H 5  G 13 ,÷òîâëå÷åò 5  13;ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  L 2 (27), L 3 (3), Sz (8)è L 2 (64). Åñëè K H  = L 2 (25),òî L 2 (25)  G H  Aut( L 2 (25)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 9  3 2  7èëè j H j =2 7  3 2  7.Âñèëóíèëüïîòåíòíîñòè OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 813 ãðóïïû H áóäåò2  7â € ( G );ïðîòèâîðå÷èå. Åñëè K H  = 2 F 4 (2) 0 ,òî 2 F 4 (2) 0  G H  Aut( 2 F 4 (2) 0 ),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2  7èëè j H j =7.Ïóñòü H 7 2 Syl 7 ( H )è G 13 2 Syl 13 ( G ). Òîãäà H 7 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäàñëåäóåò,÷òî H 7 E G . Òàêèìîáðàçîì, A = H 7  G 13 ïîäãðóïïàâ G .Èìååì H 7 E A ,èïîòåîðåìå Ñèëîâà G 13 E A ,àçíà÷èò, A = H 7  G 13 ,÷òîâëå÷åò7  13;ïðîòèâîðå÷èå. Åñëè K H  = G 2 (4),òîèç j G j = j G 2 (4) j âûòåêàåò,÷òî j H j =1.Ïîýòîìó G  = G 2 (4).  Ñîãëàñíî[26]èìååì  ( U 6 (2))= f 7 ; 8 ; 10 ; 11 ; 12 ; 15 ; 18 g ,îòêóäà D ( U 6 (2))= (2 ; 2 ; 2 ; 0 ; 0). Òåîðåìà4.4. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( U 6 (2))= (2 ; 2 ; 2 ; 0 ; 0) è j G j = j U 6 (2) j .Òîãäà G  = U 6 (2) . Äîêàçàòåëüñòâî. Èìååì j U 6 (2) j =2 15  3 6  5  7  11è D ( U 6 (2))=(2 ; 2 ; 2 ; 0 ; 0). Òàêæå € ( G )= f 2  3 ; 2  5 ; 3  5;7;11 g ;çíà÷èò, f 2  7 ; 2  11 ; 7  11 g\ ! ( G )= ? , Ïîýòîìóâñèëóëåììû3.1ãðóïïà G íåðàçðåøèìà. Òàêêàê t ( G )=3,òî G íåÿâëÿåòñÿíèôðîáåíèóñîâîé,íè2-ôðîáåíèóñîâîé ãðóïïîé. Ââèäóëåììû3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 - ãðóïïà.Ïîýòîìó K H  G H  Aut � K H  .Ïîñêîëüêó7 ; 11 = 2  1 ( G ),èìååì f 11 ; 7 g2  � K H  .Òåìñàìûì K H 2 S 11 è7 ; 11 j K H .Èçòàáë.1ïîëó÷àåì,÷òîãðóïïà K H èçîìîðôíà M 22 èëè U 6 (2). Åñëè K H  = M 22 ,òî M 22  G H  Aut( M 22 )ñó÷åòîìòîãî,÷òî G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 8  3 4 èëè j H j =2 7  3 4 .Ïîýòîìó G H  = M 22 èëè M 22  2. Ïóñòü H 3 2 Syl 3 ( H )è G 11 2 Syl 11 ( G ).Òîãäà H 3 char H E G âñèëóíèëüïîòåíò- íîñòè H ,îòêóäàñëåäóåò,÷òî H 3 E G .Òàêèìîáðàçîì, A = H 3  G 11 ÿâëÿåòñÿ ïîäãðóïïîéâ G .Èìååì H 3 E A ,èïîòåîðåìåÑèëîâà G 11 E A ,àçíà÷èò, A = H 3  G 11 ,îòêóäàñëåäóåò,÷òî3  11;ïðîòèâîðå÷èå. Åñëè K H  = U 6 (2),òî j G j = j U 6 (2) j âëå÷åò j H j =1.Èç € ( G )= € ( U 6 (2)) ïîëó÷àåì,÷òî G  = U 6 (2).Çíà÷èò,ãðóïïà U 6 (2)OD-õàðàêòåðèçóåìà.  Èç[25]  ( L 5 (3))= f 16 ; 18 ; 24 ; 78 ; 80 ; 104 ; 121 g ,îòêóäà D ( L 5 (3))=(3 ; 2 ; 2 ; 1 ; 0). Òåîðåìà4.5. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( L 5 (3))= (3 ; 2 ; 2 ; 1 ; 0) è j G j = j L 5 (3) j .Òîãäà G  = L 5 (3) . Äîêàçàòåëüñòâî. Èìååì j L 5 (3) j =2 9  3 10  5  11 2  13è D ( L 5 (3))=(3 ; 2 ; 2 ; 1 ; 0). Òàêæå € ( G )= f 2  3 ; 2  5 ; 2  13 ; 3  13;11 g .Òîãäà f 13  5 ; 13  11 ; 5  11 g\ ! ( G )= ? ,Ïîýòîìóâñèëóëåììû3.1 G íåðàçðåøèìà,îòêóäàñëåäóåò,÷òî G íåÿâëÿåòñÿ 2-ôðîáåíèóñîâîéãðóïïîéïîëåììå3.2(c). Ïðåäïîëîæèì,÷òî G íåðàçðåøèìàÿôðîáåíèóñîâàãðóïïàñôðîáåíèóñî- âûìäîïîëíåíèåì H èôðîáåíèóñîâûìÿäðîì K .Èñïîëüçóÿòåæåîáîçíà÷åíèÿ. ÷òîâëåììå3.3(c),ïîëó÷àåì,÷òî13 2  ( Z ),îòêóäàñëåäóåò,÷òîâ H 0 èìååòñÿ ýëåìåíòïîðÿäêà13  5;ïðîòèâîðå÷èå.Çíà÷èò, G íåÿâëÿåòñÿíèôðîáåíèóñîâîé, íè2-ôðîáåíèóñîâîéãðóïïîé. Âñèëóëåììû3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 -ãðóïïà.Ñëåäîâàòåëüíî, K H  G H  Aut � K H  .Ïîñêîëüêó11 = 2  1 ( G ),èìååì 11 2  � K H  .Ïîýòîìó K H 2 S 13 è11 j K H .Èçòàáë.1èòîãîôàêòà,÷òî7 - j G j , ïîëó÷àåì,÷òîãðóïïà K H èçîìîðôíà L 2 (11), L 3 (3), L 5 (3), L 4 (3), M 11 èëè M 12 . 814 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ Åñëè K H  = L 2 (11),òî L 2 (11)  G H  Aut( L 2 (11)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 7  3 9  11  13èëè j H j =2 6  3 9  11  13.Âñèëóíèëüïîòåíòíîñòè H èìååì13  11â € ( G );ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  M 11 , M 12 . Åñëè K H  = L 3 (3),òî L 3 (3)  G H  Aut( L 3 (3)),òàêêàê G H  Aut � K H  .Ñëåäî- âàòåëüíî, j H j =2 5  3 7  5  11 2 èëè j H j =2 4  3 7  5  11 2 .Âñèëóíèëüïîòåíòíîñòè H èìååì5  11â € ( G );ïðîòèâîðå÷èå. Åñëè K H  = L 4 (3),òî L 4 (3)  G H  Aut( L 4 (3)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 2  3 4  11 2 èëè j H j =3 4  11 2 .Ââèäóíèëüïîòåíòíîñòè H áóäåò3  11â € ( G );ïðîòèâîðå÷èå. Ïîýòîìó K H  = L 5 (3)èòîãäàèç j G j = j L 5 (3) j âûòåêàåò,÷òî j H j =1.Çíà÷èò, G  = L 5 (3).Ñëåäîâàòåëüíî,ãðóïïà L 5 (3)OD-õàðàêòåðèçóåìà.  Ñîãëàñíî[26]èìååì  ( O + 8 (3))= f 8 ; 12 ; 13 ; 14 ; 15 ; 18 ; 20 g ,îòêóäà D ( O + 8 (3))= (3 ; 2 ; 2 ; 1 ; 0). Òåîðåìà4.6. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( O + 8 (3))= (3 ; 2 ; 2 ; 1 ; 0) è j G j = j O + 8 (3) j .Òîãäà G  = O + 8 (3) . Äîêàçàòåëüñòâî. Èìååì j O + 8 (3) j =2 12  3 12  5 2  7  13è D ( O + 8 (3))= (3 ; 2 ; 2 ; 1 ; 0).Êðîìåòîãî, € ( G )= f 2  3 ; 2  5 ; 2  7 ; 3  5;13 g ,òîãäà f 13  5 ; 13  7 ; 5  7 g\ ! ( G )= ? .Ïîýòîìóïîëåììå3.1 G íåðàçðåøèìà,îòêóäàñëåäóåò, ÷òî G íåÿâëÿåòñÿ2-ôðîáåíèóñîâîéãðóïïîéâñèëóëåììû3.2(c). Ïðåäïîëîæèì,÷òî G íåðàçðåøèìàÿôðîáåíèóñîâàãðóïïàñôðîáåíè- óñîâûìäîïîëíåíèåì H èôðîáåíèóñîâûìÿäðîì K .Èñïîëüçóÿîáîçíà÷åíèÿ ëåììû3.3(c),ïîëó÷àåì,÷òî7 2  ( Z ).Ñëåäîâàòåëüíî,â H 0 åñòüýëåìåíò ïîðÿäêà7  5;ïðîòèâîðå÷èå.Çíà÷èò, G íåÿâëÿåòñÿíèôðîáåíèóñîâîé,íè2- ôðîáåíèóñîâîéãðóïïîé. Âñèëóëåììû3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H  íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøèìàÿ  1 -ãðóïïà.Ïîýòîìó K H  G H  Aut � K H  .Ïîñêîëüêó13 = 2  1 ( G ),èìååì13 2  � K H  .Ïîýòîìó K H 2 S 13 è13 j K H .Èçòàáë.1èòîãîôàêòà,÷òî11 = 2  ( G ), ïîëó÷àåì,÷òîãðóïïà K H èçîìîðôíà L 2 (13), L 2 (25), L 2 (27), Sz (8), L 3 (3), U 3 (4), L 4 (3), L 2 (64), 2 F 4 (2) 0 , L 3 (9), G 2 (3), S 6 (3), O + 8 (3), O 7 (3)èëè G 2 (4). Åñëè K H  = L 2 (25),òî L 2 (25)  G H  Aut( L 2 (25)),òàêêàê G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 9  3 11  7, j H j =2 8  3 11  7èëè j H j =2 7  3 11  7.Âñèëó íèëüïîòåíòíîñòè H ,3  7â € ( G );ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  L 3 (3), U 3 (4), L 4 (3), 2 F 4 (2) 0 . Åñëè K H  = G 2 (3),òî G 2 (3)  G H  Aut( G 2 (3)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 6  3 6  5 2 èëè j H j =2 5  3 6  5 2 .Ïîýòîìó G H  = G 2 (3)èëè G H  = G 2 (3)  2,ãäå  ( H ) f 2 ; 3 ; 5 g .Ïóñòü H 5 2 Syl 5 ( H )è G 13 2 Syl 13 ( G ).Òîãäà H 5 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäàñëåäóåò,÷òî H 5 E G .Òàêèì îáðàçîì, A = H 5  G 13 åñòüïîäãðóïïàâ G .Èìååì H 5 E A ,èïîòåîðåìåÑèëîâà G 13 E A ,îòêóäà A = H 5  G 13 ,÷òîâëå÷åò5  13;ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  L 2 (13), L 2 (27), L 2 (64), L 3 (9), Sz (8)èëè S 6 (3). Åñëè K H  = G 2 (4),òî G 2 (4)  G H  Aut( G 2 (4)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =3 9 .Ïîýòîìó G H  G 2 (4)  2.Ïóñòü H 3 2 Syl 3 ( H )è G 7 2 Syl 7 ( G ).Òîãäà H 3 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäàñëåäóåò, ÷òî H 3 E G .Ñäðóãîéñòîðîíû,ñèëîâñêàÿ7-ïîäãðóïïàãðóïïû G äåéñòâóåòíà OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 815 H 3 áåçíåïîäâèæíûõòî÷åê,ïîòîìó÷òî3  7.Òàêèìîáðàçîì,äîëæíîáûòü 7 j (3 9 � 1);ïðîòèâîðå÷èå. Íàêîíåö, K H  = O + 8 (3),èèç j G j = j O + 8 (3) j ñëåäóåò,÷òî j H j =1.Çíà÷èò, G  = O + 8 (3).Ïîýòîìóãðóïïà O + 8 (3)OD-õàðàêòåðèçóåìà.  Ñîãëàñíî[9]  ( S 4 (8))= f 4 ; 18 ; 14 ; 63 ; 65 g ,îòêóäà D ( S 4 (8))=(2 ; 2 ; 1 ; 2 ; 1). Òåîðåìà4.7. Ïóñòü G êîíå÷íàÿãðóïïàòàêàÿ,÷òî D ( G )= D ( S 4 (8))= (2 ; 2 ; 1 ; 2 ; 1) è j G j = j S 4 (8) j .Òîãäà G  = S 4 (8) . Äîêàçàòåëüñòâî. Èìååì j S 4 (8) j =2 12  3 4  5  7 2  13è D ( S 4 (8))=(2 ; 2 ; 1 ; 2 ; 1). Òàêæåèìååì € ( G )= f 2  3 ; 2  7 ; 3  7;5  13 g .Òàêèìîáðàçîì, G èìååò íåñâÿçíûéãðàôïðîñòûõ÷èñåëòàêîé,÷òî s ( G )=2.Ïîêàæåì,÷òî G íåÿâëÿ- åòñÿíèôðîáåíèóñîâîé,íè2-ôðîáåíèóñîâîéãðóïïîé.Åñëè G ôðîáåíèóñîâà ãðóïïà,òîïîëåììå3.4 G = KC ñôðîáåíèóñîâûìÿäðîì K èôðîáåíèóñî- âûìäîïîëíåíèåì C ñîñâÿçíûìèêîìïîíåíòàìè € ( K )è € ( C ). € ( K )ãðàôñ âåðøèíîé f 5 ; 13 g ,à € ( C )ãðàôñâåðøèíàìè f 2 ; 3 ; 7 g .Âñèëóëåììû3.3(b) j K jj ( j C j� 1).Òàêêàê j K j =5  13è j C j =2 12  3 4  7 2 ,òî(5  13) - (2 10  3 2  5 2 � 1);ïðî- òèâîðå÷èå.Åñëè G 2-ôðîáåíèóñîâàãðóïïà,òîñóùåñòâóåòíîðìàëüíûéðÿä 1 /H/K/G òàêîé,÷òî K è G=H ôðîáåíèóñîâûãðóïïûñÿäðàìè H è K=H ñî- îòâåòñòâåííî.Ïîëåììå3.2(a)èìååì T ( G )= f  1 ( G )=  ( H ) [  ( G=K ) ; 2 ( G )=  ( K=H ) g .Ïîýòîìó j K=H j =5  13.Äàëåå,ïîëåììå3.2(b)èëåììå3.6èìååì G=K  Aut( K=H )  = ( Z 12  Z 4 )  2!,àçíà÷èò, j G=K jj 2 5  3,îòêóäàñëåäóåò,÷òî f 5 ; 7 ; 13 g  ( K ).Îòñþäà7 2  ( H ).Ïóñòü H 7 2 Syl 7 ( H )è G 13 2 Syl 13 ( G ). Òîãäà H 7 char H E G .Âñèëóíèëüïîòåíòíîñòè H ïîëó÷àåì,÷òî H 7 /G è H 7 äåéñòâóåòíà G 13 áåçíåïîäâèæíûõòî÷åê,ïîñêîëüêó7  13â € ( G ).Òåìñàìûì äîëæíîáûòü j G 13 jj ( j H 7 j� 1),ò.å.13 j (7 i � 1), i =1 ; 2;ïðîòèâîðå÷èå. Ñîãëàñíîëåììå3.5(b) G èìååòíîðìàëüíûéðÿä1 E H/K E G òàêîé,÷òî H íèëüïîòåíòíàÿ  1 -ãðóïïà, K H íåàáåëåâàïðîñòàÿãðóïïàè G K ðàçðåøè- ìàÿ  1 -ãðóïïà.Ñëåäîâàòåëüíî, K H  G H  Aut � K H  .Òàêêàê5 ; 13 = 2  1 ( G ),òî f 13 ; 5 g  � K H  .Ïîýòîìó K H 2 S 13 è5 ; 13 j K H .Èçòàáë.1èòîãîôàêòà,÷òî 11 = 2  ( G ),ïîëó÷àåì,÷òîãðóïïà K H èçîìîðôíà L 2 (13), L 2 (27), L 2 (64), Sz (8), 3 D 4 (2)èëè S 4 (8). Åñëè K H  = L 2 (13),òî L 2 (13)  G H  Aut( L 2 (13)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 10  3 3  5  7èëè j H j =2 9  3 3  5  7.Âñèëóíèëüïîòåíòíîñòè H èìååì5  7â € ( G );ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  L 2 (27). Åñëè K H  = L 2 (64),òî L 2 (64)  G H  Aut( L 2 (64)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =2 6  3 2  7, j H j =2 6  3  7, j H j =2 5  3 2  7èëè j H j =2 5  3  7. Ïîýòîìó G H  = L 2 (64)èëè G H  = L 2 (64)  6,ãäå  ( H ) f 2 ; 3 ; 7 g .Ïóñòü H 7 2 Syl 7 ( H ) è G 13 2 Syl 13 ( G ).Òîãäà H 7 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäà ñëåäóåò,÷òî H 7 E G .Òàêèìîáðàçîì, A = H 7  G 13 åñòüïîäãðóïïàâ G .Èìååì H 7 E A ,èïîòåîðåìåÑèëîâà G 13 E A ,îòêóäà A = H 7  G 13 ,÷òîâëå÷åò7  13; ïðîòèâîðå÷èå. Àíàëîãè÷íîäîêàçûâàåòñÿ,÷òî K H  Sz (8). Åñëè K H  = 3 D 4 (2),òî 3 D 4 (2)  G H  Aut( 3 D 4 (2)),ïîñêîëüêó G H  Aut � K H  . Ñëåäîâàòåëüíî, j H j =5.Ïîýòîìó G H  = 3 D 4 (2).Ïóñòü H 5 2 Syl 5 ( H )è G 7 2 Syl 7 ( G ).Òîãäà H 5 char H E G âñèëóíèëüïîòåíòíîñòè H ,îòêóäàñëåäóåò,÷òî H 5 E G .Òåìñàìûì A = H 5  G 7 ÿâëÿåòñÿïîäãðóïïîéâ G .Èìååì H 5 E A ,èïî òåîðåìåÑèëîâà G 7 E A ,îòêóäà A = H 5  G 7 ,÷òîâëå÷åò5  7;ïðîòèâîðå÷èå. 816 Ï.Íîñðàòïóð,Ì.Ð.Äàðàôøåõ Íàêîíåö, K H  = S 4 (8),àçíà÷èò, j G j = j S 4 (8) j .Ñëåäîâàòåëüíî, j H j =1. Ïîýòîìóòàêêàê € ( G )= € ( S 4 (8)),òî G  = S 4 (8).Çíà÷èò,ãðóïïà S 4 (8)OD-õà- ðàêòåðèçóåìà.  ËÈÒÅÐÀÒÓÐÀ 1. ÂàñèëüåâÀ.Â. Îðàñïîçíàâàåìîñòèâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï,ïðîñòûå äåëèòåëèïîðÿäêîâêîòîðûõíåïðåâîñõîäÿò13//Ñèá.ìàò.æóðí..2005.Ò.46,2. Ñ.315324. 2. ZavarnitsineA.V. Finitesimplegroupswithnarrowprimespectrum//Sib.Electron.Math. Rep..2009.V.6.P.112. 3. MoghaddamfarA.R.,ZokayiA.R.,DarafshehM.R. Acharacterizationofnitesimplegroups bydegreesofverticesoftheirprimegraphs//AlgebraColloq..2005.V.12,N3.P.431442. 4. MoghaddamfarA.R.,ZokayiA.R. Recognizingnitegroupsthroughorderanddegree pattern//AlgebraColloq..2008.V.15,N3.P.449456. 5. HoseiniA.A.,MoghaddamfarA.R. Recognizingalternatinggroups A p +3 forcertainprimes p bytheirordersanddegreepatterns//Front.Math.China.2010.V.5,N3.P.541553. 6. MoghaddamfarA.R.,ZokayiA.R. OD -õàðàêòåðèçàöèÿâñåõêîíå÷íûõíåàáåëåâûõïðîñòûõãðóïï 817 26. ConwayJ.H.,CurtisR.T.,NortonS.P.,ParkerR.A.,WilsonR.A. Atlasofnitegroups. Oxford:ClarendonPress,1985. Ñòàòüÿïîñòóïèëà19îêòÿáðÿ2012ã. ParivasNosratpour(ÍîñðàòïóðÏàðèâàñ) DepartmentofMathematics,IlamBranch, IslamicAzadUniversity,Ilam,Iran. [email protected] MohammadRezaDarafsheh(ÄàðàôøåõÌîõàììåäÐåçà) SchoolofMathematics,StatisticsandComputerScience, CollegeofScience,UniversityofTehran,Tehran,Iran [email protected]

Приложенные файлы

  • pdf 89244833
    Размер файла: 510 kB Загрузок: 0

Добавить комментарий