ÓÕÍÏÐÔÉÊÅÓ ÁÐÁÍÔÇÓÅÉÓ ÈÅÌÁÔÙÍ

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1 ÐáíåðéóôÞìéï ÊñÞôçò, ÔìÞìá Ìáèçìáôéêþí Èåùñßá ÏìÜäùí (M 22) ÅîÝôáóç Éáíïõáñßïõ 2006 ÅîåôáóôÞò: ÄçìÞôñéïò ÍôáÞò ÓÕÍÏÐÔÉÊÅÓ ÁÐÁÍÔÇÓÅÉÓ ÈÅÌÁÔÙÍ ÈÅÌÁ ï (i) Âë. èåùñþìáôá êáé (áðü ôéò óçìåéþóåéò ðáñáäüóåùí ôïý äéäüóêïíôïò). (ii) Âë. Üóêçóç , èåþñçìá , ëþììá êáé èåþñçìá ÈÅÌÁ 2ï (i) Âë. ëþììá (ii) Âë. èåþñçìá ÈÅÌÁ 3ï Âë. ëþììá 4.3., èåùñþìáôá 4.3.2, 4.3.2, åöáñìïãýò. ÈÅÌÁ 4ï Âë. ëþììá 5.. êáé ðñüôáóç ÈÅÌÁ 5ï (i) Âë. èåþñçìá (ii) Âë. ðüñéóìá êáé ðñüôáóç ÈÅÌÁ 6ï (i) Êáô' áñ Üò äéáðéóôþíïõìå üôé ôï Z p áðïôåëåß õðïïìüäá ôþò ïìüäáò (Q/Z, +) ìýóù ôïý (ii) ôþò ðñïôüóåùò ÐñÜãìáôé ãéá ïéáäþðïôå, Z êáé i, j N 0 Ý ïõìå á) + = + = p j +p i Z p i p j p i p j p i+j p (êëåéóôüôçôá ôþò +''), 0 â) 0+Z = Z p i p (ôï ïõäýôåñï óôïé åßï ôþò Q/Z ðåñéý åôáé óôçí Z p ), êáé ã) åüí Z p i p,ôüôåðñïöáíþò Z p i p. Åí óõíå åßá, ðáñáôçñïýìå üôé ôï õðïóýíïëï p i ª i N0 ôþò Zp åßíáé Üðåéñï (ïðüôå êáé crd(z p )= ), äéüôé áðïôåëåßôáé áðü æåýãç óáöþò äéáêåêñéìýíùí óôïé åßùí. ÐñÜãìáôé åüí i, j N 0 êáé õðïèýóïõìå üôé p i = p j, ôüôå ν Z : p j = p i (νp j +), êüôé ôï ïðïßï åßíáé áäýíáôïí üôáí ν 6= 0. ÁëëÜ ν =0éóïäõíáìåß ìå ôï üôé i = j. (ii) Ðñïöáíþò, x = Z p i p Ý ïõìå x = = (p i ) p n ª p n i N0. (iii) Ãéá êüèå óôïé åßï x = ôþò Z p i p éó ýåé p i x = p i = = Z =0 p i Zp, ïðüôå ord(x) p i (âë. ðñüôáóç 2.3.8), ðïõ óçìáßíåé üôé ord(x) =p n ãéá êüðïéïí n {0,,...,i}. (iv) óôù H ìéá õðïïìüäá ôþò Z p, ãéáôçíïðïßáõðüñ åéüíùöñüãìáôïýóõíüëïõôùíôüîåùíôùí óôïé åßùí ôçò. óôù := mx{ ord(h) h H}. ÅðéëÝãïõìå Ýíá óôïé åßï x ôþò H ìå ìýãéóôç ôüîç. Áõôü èá åßíáé ôþò ìïñöþò x = ìå ìêä(, )=.[ÅÜí p j ãéá êüðïéïí j {,...,k}, ôüôå j x =0, êüôé ðïõ èá áíôýêåéôï ðñïò ôçí åðéëïãþ ôïý x.] Ùò åê ôïýôïõ, õðüñ ïõí r, s Z, ôýôïéïé þóôå íá éó ýåé r + s =. ÅðïìÝíùò, ãéá ïéïíäþðïôå n Z Ý ïõìå n p k = n = nrpk +ns = ns + nr = ns p k = ns = nsx H,

2 ïðüôå p k H. Áðü ôçí Üëëç ìåñéü, êüèå óôïé åßï h ôþò ïìüäáò H ãñüöåôáé õðü ôçí ìïñöþ h =, üðïõ Z, ìêä(, p j ) = êáé j {,...,k} (äéüôé åî ïñéóìïý ord(h) k). ÊáôÜ p j óõíýðåéáí, h = pk j = j p k p k. Ôåëéêþò, H = p k. ÅðéðñïóèÝôùò, äõíüìåé ôþò ðáñáôçñþóåùò 2.3.2, Ý ïõìå H = (åðåéäþ ord(p k )= ). (v) Ãéá ïéáäþðïôå (êõêëéêþ) õðïïìüäá H üðùò óôï (iv) (ôüîåùò H = )ïñßæïõìåôçíáðåéêüíéóç f : Z p Z p, f := pk, p i p i p i Z p. Ç f åßíáé åðéññéðôéêþ, äéüôé ãéá êüèå p i Z p Ý ïõìå f p i+k =. p i ÅðåéäÞ ³ f + p i p j ³³ = f = f + p ³ i p j +p i p i+j p j = pk (p j +p i ) p i+j êáé f + f p i p j = ³ p k p i = pk (p j +p i ) p i+j + p k p j ãéá ïéáäþðïôå, Z êáé i, j N 0, ç f åßíáé åðéìïñöéóìüò ïìüäùí ìå ðõñþíá ôïõ ôïí n o Ker(f) = Z p i p i Z ½ ¾ = Z p i p = λpi, ãéá êüðïéïí λ Z pk D E = p k = H. (iv) Åöáñìüæïíôáò ôï ï èåþñçìá éóïìïñöéóìþí ïìüäùí (3.4.) ëáìâüíïõìå Z p /H Z p. (Óçìåßùóç: ÌÝóù áõôïý ôïý éóïìïñöéóìïý ìðïñåß êáíåßò íá äþóåé ìéá åíáëëáêôéêþ áðüäåéîç ãéá ôï üôé crd(z p )=. ÅÜí ßó õå crd(z p ) = Z p <, ôüôå èá êáôáëþãáìå óå áíôßöáóç ëüãù ôþò Z p /H = Z p : H êáé ôïý èåùñþìáôïò ) ÈÅÌÁ 7ï óôù G ìéá ðåðåñáóìýíç ïìüäá. ÕðïèÝôïõìå üôé ï ðëçèéêüò áñéèìüò ôùí êëüóåùí óõæõãßáò ôçò éóïýôáé ìå n êáé üôé Z(G) = m. Ç åîßóùóç êëüóåùí óõæõãßáò (âë. ðñüôáóç êáé ðáñáôþñçóç 4..7) ãñüöåôáé ùò åîþò: G = m + nx j=m+ G : C G(x j), (~) üðïõ C G (x j ) ç êåíôñïðïéïýóá ôïý {x j } êáé ôá x j óôïé åßá åðéëå èýíôá ùò åêðñüóùðïé ôùí êëüóåùí óõæõãßáò ìå ðëçèéêü áñéèìü >. ÕðïèÝôïõìå üôé n 3 êáé äéáêñßíïõìå ðåñéðôþóåéò: (i) n =. Óå áõôþí ôçí ðåñßðôùóç, åðåéäþ m, Ý ïõìå m = n =, Þôïé üôé ç G åßíáé ôåôñéììýíç. (ii) n =2. ÅÜí m = n =2, ôüôå Z(G) =G êáé ç G åßíáé áâåëéáíþ ôüîåùò 2, Þôïé G Z 2 (ðñâë. ðüñéóìá 2.4.4). Ôï åíäå üìåíï íá Ý ïõìå m =áðïêëåßåôáé ùò áêïëïýèùò: ÅÜí åß áìå m =, ôüôå ç (~) èá åãñüöåôï G =+ G : C G (x 2 ) =+ G C = G(x 2) C + G(x 2) G =. ÅðåéäÞ G 2, èá åß áìå C G(x 2) 2. Ðñïöáíþò, C G(x 2) 6=. ÁëëÜ C G(x 2) =2èá óþìáéíå üôé C G (x 2 ) Z 2 G, ðñüãìá Üôïðï (áöïý õðåôýèç üôé m =). 2

3 (iii) n =3. ÅÜí m = n =3, ôüôå Z(G) =G êáé ç G åßíáé áâåëéáíþôüîåùò 3, Þôïé G Z 3 (ðñâë. ðüñéóìá 2.4.4). ÅÜí m =2, ôüôå ç (~)ãñüöåôáé G =2+ G : C G (x 3 ) = C + 2 G(x 2) G =. ÅðåéäÞ ç G äåí åßíáé áâåëéáíþ (ðñâë. ðüñéóìá êáé èåùñþìáôá 4.4. êáé 4.4.2, Þ áðåõèåßáò ôïí ðßíáêá ôáîéíïìþóåùò ôùí ïìüäùí ìéêñþò ôüîåùò), G 6 = 2 G. ñá CG(x2) 3, Þôïé 3 2 C G (x 2 ) =, ðñüãìá áäýíáôï! ÅÜí, áðü ôçí Üëëç ìåñéü, m =, ç(~)ãñüöåôáé G =+ G : C G(x 2) + G : C G(x 3) = C G (x 2 ) + C G (x 3 ) + G =. Ãéá íá áðëïðïéþóïõìå ôïí óõìâïëéóìü áò èýóïõìå := C G (x 2 ),:= C G (x 3 ),c:= G. Äß ùò âëüâç ôþò ãåíéêüôçôáò õðïèýôïõìå üôé. ÅðåéäÞ ç G äåí åßíáé áâåëéáíþ, c 6. Áíáæçôïýìå ëïéðüí ôéò ëýóåéò ôþò åîéóþóåùò + + =, üðïõ 2 <c, c 6. c ( ) Çðåñßðôùóç = áðïêëåßåôáé, äéüôé ôüôå èá åß áìå 2 + c = = 2 = = =2, 5 Þôïé ôéìýò ðïõ äåí éêáíïðïéïýí ôçí ( ). ñá 2 <<c,c 6. ÅÜí =2êáé =3, ëáìâüíïõìå c =6. ÅÜí =2êáé >3, ôüôå c =< = c<6, c ðñüãìá åî õðïèýóåùò áäýíáôï. ÅðéðñïóèÝôùò, åüí 3 êáé 4, ôüôå + + c = c = c 2 = c {, 2}, 5 êüôé ðïõ åßíáé êáé ðüëé åî õðïèýóåùò áðïêëåéüìåíï. Ùò åê ôïýôïõ, G =6= G åßíáé éóüìïñöç åßôå ìå ôçí Z 6 åßôå ìå ôçí D 3 (åðß ôç âüóåé ôïý èåùñþìáôïò 4.4.). ÅðåéäÞ üìùò ç G õðåôýèç ðùò åßíáé ìç áâåëéáíþ, Ý ïõìå êáô' áíüãêçí G D 3. Ôåëéêü óõìðýñáóìá: ÊÜèå ðåðåñáóìýíç ïìüäá ðïõ äéáèýôåé ôï ðïëý 3 äéáöïñåôéêýò êëüóåéò óõæõãßáò ïöåßëåé íá åßíáé åßôå ôåôñéììýíç åßôå éóüìïñöç ìå ìßá åê ôùí ïìüäùí: Z 2, Z 3,D 3. ÈÅÌÁ 8ï ÕðïèÝôïõìå üôé ïé äïèåßóåò Üíôñåò êåßíôáé åðß åíüò åðéðýäïõ ó çìáôßæïíôáò êïñõöýò êáíïíéêïý åîáãþíïõ. óôù X ôï óýíïëï üëùí ôùí äõíáôþí ó çìáôéóìþí êáíïíéêþí åîáãþíùí áõôïý ôïý åßäïõò. Ãéá íá ðñïóäéïñßóïõìå ôá óôïé åßá ôïý X ìðïñïýìå íïåñü íá öáíôáóèïýìå üôé äéáèýôïõìå 6 Üóðñåò Üíôñåò êáé üôé êáôáãñüöïõìå üëïõò ôïõò äõíáôïýò ôñüðïõò «ìáõñßóìáôïò» ôñéþí åî áõôþí. ÅðïìÝíùò, ï ðëçèéêüò áñéèìüò ôïý X éóïýôáé ìå ôïí óõíäõáóìü crd(x) = 6 3 =20ôùí 6 êïñõöþí áíü 3. Ç«ïðôéêïðïßçóç» ôùí êïñõöþí ôùí 20 êáíïíéêþí åîáãþíùí óôï áñôß ìáò ìå Üóðñåò êáé ìáýñåò ìðáëßôóåò (üðùò óôï Ó Þìá ) åßíáé áñêïýíôùò âïçèçôéêþ. Ùò ãíùóôüí, ç ÐåñÓõìì(P ) áðïôåëåßôáé áðü ôéò 6 óôñïöýò g k := R k, ðåñß ôï êýíôñï ôïý P êáôü ãùíßá 2(k )π 6,k=, 2, 3, 4, 5, 6, (üðïõ R çóôñïöþêáôüãùíßá 2π 6 ). Ç Óõìì(P ) Ý åé ôüîç 2 êáé áðïôåëåßôáé áðü ôéò 6 óôñïöýò { g k k 6}, áðü ôïõò 3 êáôïðôñéóìïýò g 7,g 8,g 9 ùò ðñïò ôéò 3 äéáãùíßïõò ôïý P, êáé áðü ôïõò 3 êáôïðôñéóìïýò g 0,g,g 2 ùò ðñïò ôéò 3 åõèåßåò ôéò äéåñ üìåíåò áðü ôá ìåóïóçìåßá ôùí (3 æåõãþí) áíôéêåéìýíùí ðëåõñþí ôïý P. Ãéá íá ëýóïõìå ôï ðñüâëçìá áñêåß íá åöáñìüóïõìå ôïí ôýðï êáôáìåôñþóåùò ôùí ôñï éþí ôþò öõóéêþò äñüóåùò ôþò G {Z 6,D 6 } åðß ôïý X, Þôïé ôïí ôýðï G X X, (g, P) 7 gp, crd({ôñï éýò ôþò äñüóåùò}) = G 3 X crd(fix G (g)) g G

4 ôïý èåùñþìáôïò 4.2.2, üðïõ Fix G (g) :=X g := { P X gp = P }. Ó Þìá Ðñïöáíþò, ç ôáõôïôéêþ áðåéêüíéóç g = Id áöþíåé üëá ôá óôïé åßá ôïý X áíáëëïßùôá. Ôï ó Þìá 2 äåß íåé ôéò êïñõöýò ôùí ìüíùí åîáãþíùí ôïý X ðïõ ìýíïõí áíáëëïßùôá êáôüðéí åöáñìïãþò ôþò óôñïöþò g 3 (êáôü 20 )ÞôÞòóôñïöÞòg 5 (êáôü 240 ). Ó Þìá 2 Ôï ó Þìá 3 äåß íåé ôéò êïñõöýò ôùí ôåóóüñùí åîáãþíùí ôïý X ðïõìýíïõí áíáëëïßùôá êáôüðéí åöáñìïãþòåíüòêáôïðôñéóìïýg j,j {7, 8, 9},ùòðñïòìéáäéáãþíéï. Ó Þìá 3 ÅßíáéåýêïëïíáäéáðéóôùèåßüôéôáëïéðÜóôïé åßáôþòäñþóáòïìüäáòäåíáöþíïõíêáíýíá åîüãùíï ôïý 4

5 X áíáëëïßùôï. Ùò åê ôïýôïõ, Ý ïõìå ôç äõíáôüôçôá êáôáñôßóåùò ôùí êáôáëüãùí Óôïé åßá äñþóáò ïìüäáò ÐëÞèïò ôùí åîáãþíùí ôïý X ðïõ ðáñáìýíïõí áíáëëïßùôá g = Id 20 g 2 0 g 3 2 g 4 0 g 5 2 g 6 0 êáé Óôïé åßá äñþóáò ïìüäáò ÐëÞèïò ôùí åîáãþíùí ôïý X ðïõ ðáñáìýíïõí áíáëëïßùôá g 7 4 g 8 4 g 9 4 g 0 0 g 0 g 2 0 ÊáôÜ óõíýðåéáí, óôçí ðåñßðôùóç (i) Ý ïõìå 6X crd(fix(g i)) = 6 6 ( ) = 24 6 = 4 êáé óôçí ðåñßðôùóç (ii) 2 X2 i= i= crd(fix(g i)) = 2 ( ) = 36 2 = 3 ïõóéùäþò äéáöïñåôéêïýò ôñüðïõò êáôáóêåõþò åíüò «óõììåôñéêïý ðåñéäýñáéïõ» êüíïíôáò ñþóç ôùí 6 äéáèýóéìùí áíôñþí. ÈÅÌÁ 9ï (i) óôù G ìéá ïìüäá ôüîåùò 665 = Óõìâïëßæïíôáòùòs 5,s 7 êáé s 9 ôïõò ðëçèéêïýò áñéèìïýò ôùí Sylow 5-, 7- êáé9-õðïïìüäùí ôþò G, áíôéóôïß ùò, ôï 3ï èåþñçìá ôïý Sylow (4.3.6) ìáò ðëçñïöïñåß üôé s s 5 (mod 5) ) = s 5 {, 5, 7, 9, 35, 95, 33, 665} s 5 (mod 5) ) = s 5 =. Áíáëüãùò áðïäåéêíýåôáé üôé s 7 = s 9 =. ÊáôÜ óõíýðåéáí, õðüñ åé ìüíïí ìßá Sylow 5-, ìüíïí ìßá Sylow 7-êáé ìüíïí ìßá Sylow 9-õðïïìÜäá ôþò G. ÂÜóåé ôïý ðïñßóìáôïò êáé ôïý ðïñßóìáôïò áõôýò ïé õðïïìüäåò (áò ôéò óõìâïëßóïõìå ùò H,H 2,H 3)åßíáéïñèüèåôåò êáé êõêëéêýò,åíþðñïöáíþòç H H 2 = H 3 H 2 = H H 3 åßíáé ôåôñéììýíç (êáèüôé ïé 5, 7, 9 åßíáé ðñþôïé). ÅðïìÝíùò, H H 2 H 3 H H 2 H 3 (ðñâë êáé 3.5.3), êáé åðåéäþ H H 2 H 3 = H H 2 H 3 =665(ðñâë (i)), Ý ïõìå G H H 2 H 3 Z5 Z 7 Z 9 (3.5.4) (Z 5 Z 7) Z 9 (3.5.8) Z 35 Z 9 (3.5.8) Z 665. (ii) óôù G ìéá ðåðåñáóìýíç ïìüäá ðïõ äéáèýôåé ìßá êáé ìüíïí ãíþóéá, ìç ôåôñéììýíç õðïïìüäá H. Ç ôüîç G ôþò G äåí ìðïñåß íá äéáéñåßôáé äéü äýï äéáêåêñéìýíùí ðñþôùí áñéèìþí p êáé q (äéüôé êáôü ôï èåþñçìá ôïý Cuchy èá õðþñ áí ôïõëü éóôïí äýï äéáêåêñéìýíåò ãíþóéåò, ìçôåôñéììýíåò õðïïìüäåò ôþò G, êüôé ðïõ èá áíôýöáóêå ðñïò ôçí õðüèåóþ ìáò). ñá G = p n, üðïõ p ðñþôïò êáé n 2(!) [ÅÜí 5

6 n =, ôüôå ç G èá Þôáí áðëþ.] ÊáôÜ ôï ï èåþñçìá ôïý Sylow (4.3.2) ç G äéáèýôåé ãíþóéåò, ìç ôåôñéììýíåò õðïïìüäåò ôüîåùò p m, ãéá êüèå m n. ÅðïìÝíùò, êáô' áíüãêçí, n =, äçëáäþ G = p 2.Ùò ãíùóôüí, ôïýôï óçìáßíåé üôé åßôå G Z p 2 åßôå G Z p Z p (âë. èåþñçìá 4.4.2). Ç äåýôåñç ðåñßðôùóç áðïêëåßåôáé, êáèüóïí ïé Z p {0} êáé {0} Z p åßíáé äýï äéáöïñåôéêýò õðïïìüäåò ôþò Z p Z p ôüîåùò p. ñá ôåëéêþò G Z p 2. ÈÅÌÁ 0ï (i) Ðñïöáíþò, ôá óôïé åßá ôþò D åßíáé ôá åîþò:... t 2,t,e,t,t 2, t 2 s, t s, s, ts, t 2 s,... ( ) Ç ðñïò ôá äåîéü ìåôáöïñü êáôü ìßá ìïíüäá t åßíáé Üðåéñçò ôüîåùò, üðùò êáé üëá ôá óôïé åßá t k,k Zr{0}, åíþ ï êáôïðôñéóìüò s Ý åé ôüîç 2. Ôá óôïé åßá ôþò ìïñöþò t k s, k Zr{0}, Þôïé ïé êáôïðôñéóìïß ùò ðñïò óçìåßá, Ý ïõí ùóáýôùò ôüîç 2, äéüôé ( (t k st k )s =(t k st k )s = =(tst)s = s 2 = e, üôáí k N, (t k s)(t k s)= (t k st k )s (t st =s) = (t k+ st k+ )s = =(t st )s = s 2 = e, üôáí k ZrN 0. ¼ìùò ôï õðïóýíïëï t k s k Z ª {e} ôùí óôïé åßùí ðåðåñáóìýíçò ôüîåùò ôþò D äåí óõãêñïôåß õðïïìüäá ôþò D, êáèüôé ãéá k, l Z, ìå k>l,ý ïõìå ìç äéáôþñçóç ôþò êëåéóôüôçôáò ôþò ðñüîåùò: (t k s)(t l s)=t k l (t l st l )s = t k l (t l st l )s = t k l ìå ord(t k l )=. (ii)ëüãùôþòìïñöþò( )ôùíóôïé åßùíôþòd Ý ïõìå D = hti ` hti s, ïðüôå [D : hti] =2. (iii) ÊáôÜ ôï (ii) êáé ôçí ðñüôáóç 3..7, hti C D. ÇóåéñÜ{e} C hti C D åßíáé ðñïöáíþò ïñèüèåôç, åíþïéðçëéêïïìüäåòôçòåßíáéáâåëéáíýò,áöïýhti/{e} hti Z êáé D / hti = D : hti =2(ðñâë. 3..5), ïðüôå D / hti Z 2 (ðñâë. ðüñéóìá 2.4.4). ñá ç D åßíáé åðéëýóéìç