• Aucun résultat trouvé

E?A?A 5?EA?AI  6A?DCEAI )C>HA  5AAIJHA 7ELAHIEJ @K EJJH=  +JA @F=A = +EJ=@AA =KHAJ 5+0

N/A
N/A
Protected

Academic year: 2022

Partager "E?A?A 5?EA?AI  6A?DCEAI )C>HA  5AAIJHA 7ELAHIEJ @K EJJH=  +JA @F=A = +EJ=@AA =KHAJ 5+0"

Copied!
35
0
0

Texte intégral

(1)

.,-/01# 2 &#3#'41# 5

6$!7#1'!48 9: !44+1;, 2 <=4# 9>?@;,#A ; <!4;9#,,#

;:1#$4 &B?<C

!"#$%& '(()

*!+,&!-,$&% .% /!-012!-$34%5 64&%5 %- 7889$341%5 ,5%80 *$,4#$99%

:"$#%&5$-1 .4 *$--,&!9; <,"% 4"$#%&5$-!$&% .% 9! /$=>,$?; +@-$2%"- AB 6,$"C!&&1 D(; &4% EB F4$55,"; F6 G)); E=G'''H I!9!$5 C%.%?

(2)
(3)

!"#$ %$& '!()*+$&

!"#$%&'$#( )$'*+,-"# ./012,3*'$"4$ $+ *55',4*+,-"#

! "#$%&'()$*&# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !+ ,#-./01.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !+! 213/.#$- '. 1&4*5(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !+!+ ,#-./01.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6 !+!7 8&*- '. )&/9&-*$*&# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! : !7 ;.1<$*&#- '=35(*><1.#). .$ %.1<$*&#- '=&%'%. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ? !7! ;.1<$*&#- 0*#<*%.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ? !7!+ @&#)$*&#- .$ <991*)<$*&#- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! + !7!7 ;.1<$*&#- '=35(*><1.#). ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! : !7!6 ;.1<$*&#- '=&%'%. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +?

!7!A B<C&%<#$D /*#&%<#$D 0&%#.- -(93%*.(%. .$ *#E3%*.(%.D .#-./01. 0&%#3 ! ! ! ! ! ! ! ! ! ! ++

!7!F G%'%. 9%&'(*$ .$ &%'%. %3)*9%&5(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +7 !7!H I<%'*#<1 '=(# .#-./01. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +7

6 7$# #+)24+2)$# *'80&),12$# 69

+! "#$%&'()$*&# ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +A +!+ B<4/<- .$ /&#&J'.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +A +!+! 8.- /<4/<- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +A +!+!+ 8.- /&#&J'.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +H +!7 8.- 4%&(9.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +K +!7! 8.- 4%&(9.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! +K +!7!+ 8.- -&(-L4%&(9.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7+

+!7!7 I&#-$%()$*&# '( 5(&$*.#$ '=(# 4%&(9. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7F +!7!6 M&/&/&%9N*-/.- '. 4%&(9.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 7:

+!7!A 8.- 4%&(9.- O#*- .$ 1=.P./91. '( 4%&(9. -Q/3$%*5(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6+

+!6 8.- <##.<(P ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6A +!6! 8.- <##.<(P ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6A +!6!+ R&(-L<##.<( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6:

+!6!7 S##.<( *#$T4%.D '*>*-.(% '. U3%& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 6K +!6!6 "'3<1 '=<##.<( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A?

+!6!A "#$.%-.)$*&#D -&//. .$ 9%&'(*$ '=*'3<(P ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A+

+!A 8.- I&%9- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! A6

"

(4)
(5)

!" "#$%&#%$!" '()*+$,-%!"

!" #$%&'()*%+'$

!"# $! %&'()*+ ,+# +"#+-.$+#/ $0(.1+% 2)*"3*2!$ +#% 4" +"#+-.$+ 54* #+ ,*##*-4$+ 2!)6(*# #(4# ,0!4%)+# "(-#

%+$# 54+ 3$!##+/ 3($$+3%*(" (4 6!-*$$+7 8+2+",!"%/ ,!"# ,0!4%)+# ,*#3*2$*"+# ,+# -!%&'-!%*54+#/ 4" +"#+-.$+

+#% %(41(4)# -4"* ,04"+ #%)43%4)+7 9" !$:;.)+ %(4% 2!)%*34$*;)+-+"%/ 4" +"#+-.$+ +#% 3(-.*"' !<+3 4"+ (4 2$4#*+4)# $(*# ,+ 3(-2(#*%*(" +% #0!22+$$+ 4"+ #%)43%4)+ !$:'.)*54+7 =(*3* 4"+ $*#%+ ,+# #%)43%4)+# !$:'.)*54+#

*-2()%!"%+# >

?(4% +" &!4% ,4 ,*!:)!--+/ (" %)(4<+ $+# #%)43%4)+# !$:'.)*54+# *-2$*54!"% 4" "(-.)+ -*"*-!$ ,+ 3("%)!*"%+#

+% +" .!#/ 3+$$+# 54* +" *-2$*54+"% 4" -!@*-4-7 A$4# (" ,+#3+",/ 2$4# $! #%)43%4)+ +#% +" 54+$54+ #()%+ #2'B 3*!$*#'+7

C+# #%)43%4)+# $+# 2$4# 3(--4"+# #("% $+# :)(42+#/ $+# !""+!4@ +% $+# 3()2#7

! ,-./-0 1% /'$'2(10

! !" #$% &'(&'%

D"%)(,4*#("# ,!"# 4" 2)+-*+) %+-2# 4"+ #%)43%4)+ !$:'.)*54+ '$'-+"%!*)+ > $+ -!:-!7 EF

(6)

"#$%&'&(% ) )* ! "!#"$%&" E $'!( )*'!" &+( )" ,+$-+#(.(+! (!."/!" T "#. 0--"&1 !" ! ". "#. !+.1 (E, T)2

! "#$"# %&' ()!* +!% &',+*'+,% #-$./,01+% .-."%!'#0,%2 3- %40&'% (%& &',+*'+,%& 5-+& &+/'0-%& (#!& -%&1+%--%&

+! %!&%"/-% %&' "+!0 (% 5-+&0%+,& -)0& %' (% (06.,%!'%& 5,)5,0.'.&2

"#$%&'&(% ) ) 3+(. A '!" -0/.(" !+! 4()" )*'! $05$0 (E, T)2 6! )(/0 7'" A "#. #$!%&' -+'/ &0 &+( T +' 7'" A "#. '!" (!)$*' #$!%&' )" E 8&+/#7'*(& !*9 0 -0# )*0$%(5:(.1 #'/ &0 &+( "! ,0'#"; #( &0 /"#./(,.(+! <

A )" &0 &+( T "#. '!" &+( )" ,+$-+#(.(+! (!."/!" )0!# A2 ='./"$"!. )(. >

A⊂E?A #.0%&" -+'/ &0 &+( T ⇔ ∀(a, b)∈A×A? aT b∈A

7! 5%+' ()!* ,%"#,1+%, 1+% (0,% 1+8+!% -)0 T %&' +!% -)0 (% *)"5)&0'0)! 0!'%,!% (#!& +! %!&%"/-% E .1+09#+' : (0,% 1+% E %&' &'#/-% 5)+, *%''% -)02 ;! ,%9#!*<%= &0 A %&' +! &)+&>%!&%"/-% (% E= 0- !8%&' 5#&

*%,'#0! 1+%A &)0' &'#/-% 5)+, -# -)0T2 ?0!&0N!8%&' 5#& &'#/-% 5)+, -# &)+&',#*'0)!= "#0&Z-8%&'2 @% "A"%=

Z !8%&' 5#& &'#/-% 5)+, -80!9%,&%= "#0& Q -8%&'2

"#$%&'&(% ) )+ 3(? -+'/ .+'. ,+'-&" (x, y) )*1&1$"!.# )" E? +! 0 xT y =yT x? &0 &+( T "#. )(." +, -.

$!$*/'2 (E, T) "#. )(. +, -$!$*02 ,-./012- ) )*

B C%& "#$"#& (N,+)= (Z,+)= (Z,×) &)!' *)""+'#'0D&2 E% !8%&' 5#& -% *#& (% (Z,−) F 3−2 = 1 #-),&

1+%2−3 =−12 @% "A"%=(Q− {0}, /)= %!&%"/-% (%& !)"/,%& ,#'0)!!%-& !)! !+-& "+!0 (% -# (090&0)!

+&+%--% !8%&' 5#& *)""+'#'0D F3/26= 2/32

B C),&1+8+!% -)0 !8%&' 5#& *)""+'#'09%= *%,'#0!& .-."%!'& 5%+9%!' *%5%!(#!' G*)""+'%,H F )! 5#,-%

(8.-."%!'& 3-0.2'/45-62 3- &+I' 5)+, &8%! *)!9#0!*,% (% *)!&0(.,%, -# -)0 G)H (% *)"5)&0'0)! (%&

#55-0*#'0)!& %' -%& D)!*'0)!&f :x7→3x+ 2%'g:x7→2x+ 12

B @#!& +! "#$"# #&&)*0#'0D(E, T)= +! .-."%!' x5%,"+'#/-% #9%* (%+4 .-."%!'&a%'b %&' 5%,"+'#/-%

#9%* -% *)"5)&. aT b2

J0 -# -)0 T !8%&' 5#& #&&)*0#'09%= -% ,.&+-'#' 5%+' !% 5-+& A',% 9,#0 F (#!& (Z, T) #9%* aT b =a2−2b=3

%' −5 *)""+'%!' ()!*3 *)""+'% #9%*3 %' −5 "#0&3 !% *)""+'% 5#& #9%*19 = 3T(−5)K %! %6%' 3T19 =−29 %'19T3 = 3552

"#$%&'&(% ) )7 ! 1&1$"!. e )" E 41/(@0!. xT e = eT x = x "#. )(. 1'-$)' -+'/ &0 &+( T2 A" $05$0 (E, T) $'!( )" ,". 1&1$"!. !"'./" "#. )(. -1*02)' 8+' -0/B+(# -1*$!*)';2

89-.35- ) )*

B @#!& -%& "#$"#& (N,+) %' (Z,+)= G0H %&' !%+',%= *8%&'>:>(0,% 1+% 5)+, ')+' %!'0%, !#'+,%- )+ ')+'

%!'0%, ,%-#'0Dn=n+ 0 = 0 +n=n2

B @#!& (Z,−)=0 !8%&' !%+',% 1+8: (,)0'% Fx−0 =x "#0&0−x=−x2 B @#!& N%'Z= G1 H %&' !%+',% 5)+, -# "+-'05-0*#'0)!2

B @#!& M2(R)= %!&%"/-% (%& "#',0*%& *#,,.%& (8),(,% 2 : '%,"%& ,.%-&= -# "#',0*% I =

1 0 0 1

%&'

!%+',% 5)+, -# "+-'05-0*#'0)!2

,-./012- ) ) ! "#$"# 5%+' #("%'',% 5-+&0%+,& .-."%!'& !%+',%& (8+! G"A"%H *L'. M: (,)0'% FaT e= a= )+ : $#+*<% F eT a= aN2 O#, %4%"5-%= (#!& (Z, T) )P T (.&0$!% -# -)0 (.Q!0% 5#, aT b =a+E(b/3)-%&

%!'0%,&0=1%'2 &)!' !%+',%& : (,)0'% %' 0- !8R # 5#& (8.-."%!' !%+',% : $#+*<%2

:0(3(6&'&(% ) )* ! 1&1$"!. !"'./" < 50',C" ". < )/+(." "#. '!(7'" > ,*"#. &*1&1$"!. !"'./" )' $05$0 (E, T)2

(7)

!"#$" !"##$%$&% '"( (E, T) )*+(,,( *("- ./.+(&,% &(",0(% *1%,1&2,%e1 (,e2 3 4)"25( (, 3 *0$1,( 0(%#(26 ,17(+(&,8 9& ) *:"&( #)0, ∀x∈E;e1T x=x; 2(,,( 0(/),1$& (%, *$&2 70)1( #$"0 x=e2 2( '"1 &$"% #(0+(,

*:)<0+(0 '"( e1T e2 = e28 =:)",0( #)0, ∀x ∈ E; xT e2 = x; 2(,,( 0(/),1$& .,)&, 70)1( #$"0 x = e1; $& ) e1T e2=e18 >)0 2$&%.'"(&,;e1 =e2

"#$%&'&(% ) )* %& #' ()(*"'+ b ,"E $(!&-" aT b=bT a=b.∀a∈E. /)0!1b "1+ ,&+ ! "#$% &'()*'&$%

20#! )/ )0& T3

! ! "#$ %&'&()#$

"#$%&'&(% ) )+ %&. 20#! +0#+ +!&2)"+ (x, y, z) ,4()(*"'+1 ," E. 0' /(xT y)T z=xT(yT z). )/ )0& T "1+ ,&+"

&(()+,&%,-#3 5" */6*/ (E, T) "1+ ,&+ &(()+,&%,.3

7' *0'08," "1+ #' */6*/ /1109&/+&: "+ #'&:;!"

,-./012- ) )3

? !1 /) /$1 1&,(0&( (%, (& #/"% 2$++",),17(; &$"% *1%$&% )/$0% '"( /) %,0"2,"0( @$0+( "& .(%(45- 6(..2'/'&78

? A& 5-.&890(2:- (%, "& +)4+) )%%$21),1@8 B( +$&$C*( (%, *$&2 "& *(+1640$"#( +"&1 *:"& ./.+(&,

&(",0(8

;<-06&6- 3* D$&,0(0 '"( /:(&%(+E/( *(% (&,1(0% &),"0(/% (%, "& +$&$C*( 2$++",),1@ ,$,)/(+(&, $0*$&&.

#)0 0)##$0, )"- /$1% *:)**1,1$& (, *( +"/,1#/12),1$&8

<0!!"9+&0'

F8 B) /$1 *:)**1,1$& +(%,6(//( "&( $#.0),1$& 1&,(0&( ,(//( '"( ∀a, b∈N&$"% )G$&%a+b=c∈NH I$"%

#$"7$&% *.+$&,0(0 '"( 2:(%, E1(& /( 2)% (& %)25)&, '"( 1)##)0,1(&, 3N ,(/ '"(

a

X

i=1

1 +

b

X

i=1

1 =

a+b

X

i=1

18

=$&2 c∈N (, /:)**1,1$& (%, E1(& "&( /$1 1&,(0&( J/:(&%(+E/( (%, %,)E/( #)0 0)##$0, 3 /:)**1,1$&K (, (&

+L+( ,(+#% )%%$21),17( #"1%'"( 1#(", L,0( )**1,1$&&. 3 /"16+L+( #)0 *.M&1,1$& *)&% &:1+#$0,( '"(/

$0*0( %)&% '"( /( 0.%"/,), (& %$1, )/,.0.8

B) +"/,1#/12),1$& (%, "&( /$1 '"1 %( 2$&%,0"1, %"0 /:)**1,1$& *$&2 /) /$1 *( +"/,1#/12),1$&×(%, )"%%1 "&(

/$1 1&,(0&( (, )%%$21),17(8

N8 9& )*+(,,0) 3 #)0,10 *:121 '":1/ (%, ,0171)/ '"( /) /$1 *:)**1,1$& (%, .4)/(+(&, 2$++",),17( (, '"( /(

OPQ (& (%, /:./.+(&, &(",0( e8

B) /$1 *( +"/,1#/12),1$& (%, )"%%1 2$++",),17( (, 1/ (%, ,0171)/ '"( OFQ (& (%, /:./.+(&, &(",0( e8 R8 >$"0 2$&2/"0(; $& 0)##(//( /(% 0.%"/,),% %"17)&,%

? (N,≤,≥) (%, ,$,)/(+(&, $0*$&&. J),,(&,1$& 2(,,( &$,),1$& (%, "& #(" )E"%17(; 1/ %"<, '":1/ G )1, S"%,( "&( *(% *("- 0(/),1$&% *:$0*0(R #$"0 '"( /:(&%(+E/( %$1, ,$,)/(+(&, $0*$&&.K8

? (N,+) (,(N,×) %$&, *(% +$&$C*(% )E./1(&%8

T$&2/"%1$&;N(%, "& +$&$C*( )E./1(& ,$,)/(+(&, $0*$&&. #)0 0)##$0, )"- /$1% *:)**1,1$& (, *( +"/,1#/12),1$&8 ,-./012- ) )= U/ (%, 0)0( *:",1/1%(0 /(% +$&$C*(% 2)0 %$"7(&,; /$0%'"( &$"% &$"% ,0$"7$&% @)2( 3 "&(

%,0"2,"0( ,0$# #)"70( #$"0 #$"7$10 70)1+(&, *1%2",(0; &$"% /) #0$/$&4($&% 7(0% '"(/'"( 25$%( *( #/"% 0125(;

2$++( "& 40$"#(; $" "& )&&()" J7$10 #/"% /$1&K8

;<-.:>- ) )

? B( +)4+) (Z,+) (%, )%%$21),1@ (, "&1@V0( J/:./.+(&, &(",0( .,)&, *)&% 2( 2)% 0K *$&2 (Z,+) (%, "&

+$&$C*(8

(8)

!" #$"%& '(% )" *(% +" (Z,−), -# "."&/ (Z,−) "%& 0# 1(21( #3# 0#4567" "& #3# (%%3*4(&45 *(7/ '(7

"8"1')"/ (3−2)−5 =−4 ()37% 90" 3−(2−5) = 6, !" #$"%& '(% )" *(% #3# ')0% +" (Q− {0}, /)/ 4)

%0:& +" *3#%4+;7"7 '307 %$"# *3#<(4#*7" )$"8"1')" %04<(#& = (3/2)/5 = 3/10()37% 90" 3/(2/5) = 15/2, >( )34 +" *31'3%4&43# +"% ('')4*(&43#% "%& (%%3*4(&4<",

>" '73+04& <"*&374") ∧?@ #" '(% *3#53#+7" (<"* )" A"& B )32490"C #$"%& '(% (%%3*4(&45 = '(7 "8"1')"/ %4 (i, j, k) "%& 0#" D(%" 37&E323#()" (i∧i)∧j = 0∧k= 0 ()37% 90"i∧(i∧j) =i∧k=−j,

"#$%&'&(% ) )* !"# $" %!&%! $"'()*+ (E, T) ,-./.%+"0 "+$0*+e1 $" ./.%+"0x +#0 ,'0 !"#$%& '()*

2 3 ,*4'0+ 5 #-'/ +6'#0+x ,!"# E 0+/ 7$+ xT x =e1 2 3 &!$89+ 5 #-'/ +6'#0+x” ,!"# E 0+/ 7$+ x”T x=e:

;4*#7$+ /-./.%+"0 +#0 #<%.0*'#!=/+ 3 &!$89+ +0 3 ,*4'0+ >='/!0)*+? +0 #' x =x”1 4" ! !/4*# 5 xT x=xT x=e

!"# 8+ 8!#1 x +0x #4"0 ,'0# !"#$%&+,* @4$* /! /4' T ,+E: A" ,'0 !$##' 7$+x >*+#@+80'B+%+"0x? +#0 $"

#<%.0*'7$+ ,+ x >*+#@+80'B+%+"0x?: ;4*#7$+ /! /4' T #-'"0+*@*)0+ 84%%+ $"+ %$/0'@/'8!0'4"1 4" @!*/+ @/$0C0 ,-./.%+"0# &-.*% &()* +0 ,-&-.*% * :

+,-./01, ) )2

F(#% (Z,−)/ G "%& #"0&7" @ +734&" "& &30& "#&4"7 7")(&45 "%& %3# '73'7" "& 0#490" %H1;&7490" @ +734&", F(#% (Z,+)/ &30& "#&4"7 7")(&45x '3%%6+" 0# 0#490" %H1;&7490" = %3# (33(4#−x,

56,/7&7, 89 I3#&7"7 90" )"% )34% +$(++4&43# "& +" 10)&4')4*(&43# +" )$"#%"1D)" +"% "#&4"7% #(&07")% #$(+J 1"&&"#& '(% +" %H1;&7490",

D4**+80'4" = -84%&"J&J4) '307 )( )34 +$(++4&43# + 0# %H1;&7490" c ∈ N &") 90" ∀a ∈ N #30% (H3#% a+c =

a

X

i=1

1

!

+c =e= 0K L) "%& (%%"M &74<4() 90" '307 90" *"&&" ;2()4&; %34& %(&4%5(4&" 3# (4&

a

X

i=1

1

!

=−c⇔ a = −c 37 )"% #31D7"% #;2(&45% #$"84%&"#& '(% +(#% N, !" 904 #30% (16#" (0%%4 @ )( *3#*)0%43# 90" )( )34 +$(++4&43# +#$( '(% +" %H1;&7490" "& 90" )( )34 +" %30%&7(*&43# −#$"84%&" '(% +(#%N?)( %30%&7(*&43# ;&(#&

742307"0%"1"#& )$(++4&43# +$0# #31D7" #;2(&45C,

-84%&" &J4) '307 )( )34 +" 10)&4')4*(&43# × 0# %H1;&7490" a ∈ N &") 90" ∀a ∈ N/ 3# (4& a ×a = e = 1K F$(D37+ 4) "%& ;<4+"#& 90"a = 1

a, I(4% "8*"'&; '307 a= 1/ )" 903&4"#&1/a#$"84%&" '(% +(#%N, F3#* #30%

+"<3#% *3#*)07" 90$4) #$"84%&" '(% '307 &30& ;);1"#& +" N +" %H1;&7490"% '307 )( )34 +" 10)&4')4*(&43# "&

(4#%4 90" )( )34 +" +4<4%43# #$"84%&" '(% +(#% N,

"#$%&'&(% ) )! !"# $" %!&%! (E, T)1 $" ./.%+"0 x +#0 ,'0 %#/,)&*% >4$ &"0)&1'()*? 2 /',34* #'

@4$* 04$0 84$@/+ (a, b) ,-./.%+"0# ,+ E 0+/# 7$+ xT a = xT b1 !/4*# a = b: A" ,.E"'0 ,+ %F%+ $" ./.%+"0

%#/,)&*% 2 5%6&$*: G" ./.%+"0 +#0 ,'0 %#/,)&*% #-'/ +#0 *.&$/'+* 3 ,*4'0+ +0 3 &!$89+: H' T +#0 84%%$0!0'B+1 /+# "40'4"# ,-./.%+"0 *.&$/'+* 3 &!$89+ +0 3 ,*4'0+ 84I"8',+"0:

56,-3:, ) )8

F(#% (N,+)/ &30& ;);1"#& "%& 7;20)4"7 "& +(#%(N,×)/ &30& ;);1"#& #3# #0) "%& 7;20)4"7, F(#% (Z, T)/ (<"* aT b=|a|+b/ &30& ;);1"#& "%& 7;20)4"7 @ 2(0*E" 1(4% '(% @ +734&", +,-./01, ) )9

F(#% 0# 1(21( (%%3*4(&45/ )" *31'3%; +" +"08 ;);1"#&% 7;20)4"7% "%& 7;20)4"7, F(#% 0# 1(21( (%%3*4(&45/ &30& ;);1"#& %H1;&74%(D)" "%& 7;20)4"7,

N4 )" 1(21( #$"%& '(% (%%3*4(&45/ 74"# #$"%& (%%07;,

56,/7&7, 8* ?"4" 84**'&.C O# *3#%4+67" E )$"#%"1D)" +"% "#&4"7% #(&07")% (0 ')0% ;2(08 @ 10 "& )( )34 T +;P#4" +(#%E '(7 aT b=|a−b|?+4.;7"#*" %H1;&7490"C,

(9)

!"#$%&% '(& 0&)$ #&($%& &$ '(& $"($ *+*,&#$ &)$ )"# -%"-%& ).,*$%/'(&0 !"#$%&% '(& )&(+) 0&$10 )"#$ %*1(+/&%)0

!"#$%&% '(& T &)$ #"# 2))"3/2$/4& &# 3"#)/5*%2#$(1T2)T3&$1T(2T3)

"#$%&'&$ () 6 ! "!##$%&7 8# 3"#)/59%& 52#)Z+2 +"/T 5*:#/& -2% aT b=ab+ 30 !"#$%&% '(&T #;&)$ -2) 2))"3/2$/4&< '(& −1 &$3)"#$ %*1(+/&%) ,2/) '(& −1T3 = 00

!" #$% &'()*$%

!"!# $%& '()*+%&

=& $&%,& &)$ 5& >42%/)$& ?@=8AB 6CDCCECDFG7< +2 )$%(3$(%& &)$ 5& @(1()$/# ="(/) H@IHJK 6CLDMECDNL7<

+;2O/",2$/)2$/"# 5& @%$P(% H@K=QK 6CDGCECDMN70

*+,-'.'/- 0(01 ' %#!()* G *+, ( * +*-./* -( $ 01( * /!$ 0* "!-)!+$,$! $ ,*# * T )!(# /23(*//* /*+

24$!-*+ +($52 ,+ +! , 5&#$6&+ 7

• g1 /2 /!$ T *+, 2++!"$2,$5* 7 (xT y)T z=xT(yT z) )!(# ,!(, x8 y *, z 0* G8

• g2 /2 /!$ T )!++90* ( &/&-* , *(,#* e 7xT e=eT x=x )!(# ,!(,x 0*G8

• g3 ,!(, &/&-* , x 0* G)!++90* ( +:-&,#$3(* x )!(# /2 /!$ T 7xT x =xT x=e;

<* %#!()* G -( $ 0* +2 /!$T *+, +!(5* , !,& (G, T);

' %#!()* *+, ( -2%-2 ( $=9#* 2++!"$2,$= 02 + /*3(*/ ,!(, &/&-* , 20-*, ( +:-&,#$3(*

' %#!()* *+, ( -! !>0* 02 + /*3(*/ ,!(, &/&-* , 20-*, ( +:-&,#$3(*

"#$%&'&$ (! B"/$ G (# &#)&,R+& ,(#/ 5;(#& +"/ 5& 3",-")/$/"# /#$&%#& STT 2))"3/2$/4&< '(/ -"))95& (#

*+*,&#$ #&($%& U 5%"/$& e6/0&0 -"(% $"($x 5&G<xT e=x7 &$ $&+ '(& $"($ *+*,&#$ x -"))95& (# ).,*$%/'(&

U 5%"/$& x 6/0&0xT x =e70 !"#$%&% '(&G &)$ (# 1%"(-&0

?!##*",$! V B"/$ x ∈ G< 5& ).,*$%/'(& U 5%"/$& x0 W2% 3"#)*'(&#$ xT x &)$ (# *+*,&#$ 5& G 6+3/7< /+

-"))95& 5"#3 (# ).,*$%/'(& U 5%"/$& '(& +;"# #"$& z0 8# 2 5"#3xT((xT x)T z) =x⇒ (xT x)T xT z=x⇒ eT xT z =xT z=x< -2% 2))"3/2$/4/$* 5& +2 +"/0 8# 3",-")& &#)(/$& U 12(3P& -2%x< -"(% $%"(4&%xT(xT z) = (xT x)T z =e= xT x< 3& '(/ -&%,&$ 5;2X%,&% '(& x &)$ *12+&,&#$ (# ).,*$%/'(& U 12(3P& 5& x0 8# &#

5*5(/$ &#)(/$& '(&e&)$ 2())/ #&($%& U 12(3P&< 32% )/x&)$ 52#)G<eT x= (xT x)T x=xT(xT x) =xT e=x 6"# 2 ($/+/)* U #"(4&2( +;2))"3/2$/4/$* 5& +2 +"/ &$ +& Y2/$ '(& e&)$ (# ).,*$%/'(& U 5%"/$&70

"#$%&'&$ 23

C0 (N,+)&$(N,×) )"#$E/+) 5&) 1%"(-&) Z G0 (Z,+)&$(Z,×))"#$E/+) 5&) 1%"(-&) Z F0 (R,+)&$(R,×) )"#$E/+) 5&) 1%"(-&) Z

?!##*",$! V 8# 4*%/:& -"(% 3P23(# 5&) 3"(-+&) +&) -%"-%/*$*) 5( 1%"(-& g1 < g2 &$ g3 0 G020 W"(%(Z,+) V

∀(x, y)∈Z2<x+y∈Z< +2 +"/ &)$ 5"#3 /#$&%#& 6-2%-27<

∀x∈Z<x+ 0 = 0 +x=x0 S0T &)$ +;*+*,&#$ #&($%& 6( $=9#*7<

∀(x, y, z)∈Z3<(x+y) +z=x+ (y+z)< +2 +"/ +&)$ 2))"3/2$/4& 52#)Z 62++!"$2,$=7<

∀x∈Z<∃x =−x<x+x = 0< ,!(+ /*+ &/&-* ,+ ! , ( +:-&,#$3(*0 (Z,+)&)$ 5"#3 (# 1%"(-&0

F020 8# 5*,"#$%& 5& ,[,& '(&(R,+)&)$ (# 1%"(-&0

(10)

!" #$%&"' (

)*!* +', -+-.'%&, /'N%0$%& 1!, /0$11$,- 2,!34 056 )*7* +', -+-.'%&, /'N%0$%& 1!, /08%9'",' 2,!34 156 :*7* +', -+-.'%&, /'Z%0$%& 1!, /08%9'",' 2,!34 1 '&−156

;*7* 0 %0! 1!, /08%9'",'6

/$%# (N,+)6(N,×)6(Z,×) '&(R,×) %0$%& 1!, /' ,&"3#&3"' /' <"$31'6 #' ,$%& %-!%.$8%, /', .$%$=/',*

"#$%&'&(% )* *) ! "#$%&' ($!) *+ *$, (' -$.&$/,),$! &$//0(' ('/ -+#+-)1#,/),2%'/ /'.3*+3*'/ 4 -'**' ('

*5+((,),$! 6#'/&'-),7'.'!) *+ .%*),&*,-+),$!8 (+!/ Z6#'/&'-),7'.'!) (+!/R8 '/) (,) !!"#"$ 6#'/&'-),7'.'!)

%&'#"('") #"$89

"#$%&'&(% )* * ! "#$%&'G'/) (,) *+'",- 6(% !$. (' :,'*/ ;'!#,-< =>?@A BCDEFBCEG8 $% ).%%&# #"$

/, /+ *$, (' -$.&$/,),$! ,!)'#!' T '/) -$..%)+),7' -5'/)F4F(,#' /, xT y =yT xA ∀x, y∈G9 +,-./0- )* *1

> ?0!//8&8$% /!%, Z '& /!%, R ',& #$..3&!&89'* @' %0',& 1!, +' #!, /' +! ,$3,&"!#&8$% (2−5 6= 5−2* A!%, (Z,−)60 %0',& %'3&"' B30C /"$8&'*

> (Z,+)',& 3% <"$31' !7-+8'%6 /' .D.'(R− {0},×)* !" #$%&"'6(R− {0}, /) %' +0',& 1!, #!" +! /898,8$%

%0',& 1!, !,,$#8!&89' 2(2/3)/4 = 1/6 '&2/(3/4) = 8/35 %8 #$..3&!&89' 25/4 = 1,25'&4/5 = 0,85*

> (Z,−) %0',& 1!, 3% <"$31' ( +! ,$3,&"!#&8$% ',& 3%' +$8 /' #$.1$,8&8$% 8%&'"%' /!%, Z6 .!8, '++' %0',&

1!, !,,$#8!&89' ((2−3)−56= 2−(3−5)* @0',& #'1'%/!%& 3% .!<.!*

> E8A(R) /-,8<%' +0'%,'.7+' /', 4$%#&8$%, !F%', 78G'#&89', /' R ,3"R,$8& +0'%,'.7+' /', 4$%#&8$%, /' +! 4$".' x 7→ ax+b !9'# a %$% %3+6 (A(R),$) ',& 3% <"$31' %$% !7-+8'% 2$ ( +$8 /' #$.1$,8&8$% /',

!11+8#!&8$%,5* E8f /-,8<%'x7→2x−1'& ,8g/-,8<%'x7→3x6 $% !fog:x7→6x−1'&gof :x7→6x−3*

> E8M2(R)/-,8<%' +0'%,'.7+' /', .!&"8#', #!""-', "-'++', /0$"/"' : /' /-&'".8%!%& %$% %3+6(M2(R),×) ',& 3% <"$31' .3+&81+8#!&84 %$% !7-+8'%*

> ?0'%,'.7+' #$%,&8&3- /', H$.$&H-&8', /' "!11$"& %$% %3+ '& /', &"!%,+!&8$%, /!%, 3% 1+!%P6 .3%8 /' +!

+$8 /' #$.1$,8&8$% /', !11+8#!&8$%,6 ',& 3% <"$31' %$% #$..3&!&84* E$% -+-.'%& %'3&"' ',& +0!11+8#!&8$%

8/'%&8B3' !,,8.8+- C +! &"!%,+!&8$% /' 9'#&'3" %3+ $3 C +0H$.$&H-&8' /' "!11$"& )*

> A!%, 3% 1+!%P6 #$%,8/-"$%, 3% #!""-ABCD/' #'%&"'O* I$&$%,S0 +! ,J.-&"8' #'%&"!+' 1!" "!11$"&

C O6 S1 +! ,J.-&"8' /0!K' d1 1!,,!%& 1!" +', .8+8'3K /', #L&-, [AD] '&[BC]6 S2 +! ,J.-&"8' /0!K' d2 1!,,!%& 1!" +', .8+8'3K /', #L&-, [AB] '& [CD] '& i +0!11+8#!&8$% 8/'%&8B3' /' P* M3%8 /' +! +$8 /'

#$.1$,8&8$% /', !11+8#!&8$%,6 +0'%,'.7+'E ={i, S0, S1, S2} ',& 3% <"$31' #$..3&!&84*

+,-23&3- 41 A-#"8"' &$3, +', <"$31', 1$,,87+', C16263 $34 -+-.'%&,*

H$##'-),$! (

)* N% <"$31' C 3% -+-.'%& ',& 3% '%,'.7+'E #$%,&8&3- /03% ,'3+ -+-.'%&e6 '& +! +$8⋆',& %-#',,!8"'.'%&

/-O%8' 1!" e ⋆ e=e* P% 9-"8O' ,!%, /8F#3+&- B3'(E, ⋆) ',& 78'% 3% <"$31'*

:* E8 E #$%&8'%& /'3K -+-.'%&,6 +03% /$8& D&"' +' %'3&"' 1$3"⋆6 %$&$%, +' e'& %$&$%, +0!3&"'x* P% ! /$%#

e ⋆ e =e '&e ⋆ x =x ⋆ e= x* E8 +0$% 9'3& /' 1+3, B3' x ,$8& 8%9'",87+'6 $% /$8& %-#',,!8"'.'%& !9$8"

x ⋆ x=e6 '& $% 9-"8O' B3' (E, ⋆) ',& !+$", 3% <"$31'*

;* QG$3&$%, 3% &"$8,8R.' -+-.'%& y C %$&"' '%,'.7+'6 $% ! %-#',,!8"'.'%&e ⋆ e=e6e ⋆ x=x ⋆ e=x '&

e ⋆ y =y ⋆ e = y* P% %' 1'3& !9$8" x ⋆ y =y 2x 6=e5 %8 x ⋆ y = x 2y 6= e5 /$%# x ⋆ y =e 2+#85* P%

.$%&"' /' +! .D.' .!%8R"' B3' y ⋆ x=e* P% %' 1'3& 1!, !9$8" x ⋆ x=x 2x 6=e5 %8 x ⋆ x =e2#!"

+08%9'",' /' x ',& 3%8B3' /!%, 3% <"$31' '& $% ,!8& B308+ 9!3& y5* !" #$%,-B3'%&6x ⋆ x=y 2+#85 '& $%

.$%&"' /' +! .D.' .!%8R"' B3'y ⋆ y =x* P% ! /$%# +', "-,3+&!&, ,389!%&, (

(11)

⋆ e x y

e e x y

x x y e

y y e x

!""! #$% !&" '%!( )(! #$% *! +,$)-!.

/. 0(1(2 *3(& #! 43& *! 5)3",! 6#67!("&2 $( $'"%!(" !( 6")*%3(" "$)"!& #!& -$&&%'%#%"6& #!& 5)3",! #$%&

&)%83("!& 9

⋆ e x y z

e e x y z

x x y z e

y y z e x

z z e x y

⋆ e x y z

e e x y z

x x z e y

y y e z x

z z y x e

⋆ e x y z

e e x y z

x x e z y

y y z e x

z z y x e

⋆ e x y z

e e x y z

x x e z y

y y z x e

z z y e x

"#$%&'() *+ +! !"# $%$&'(# )*"( +,!"-' (G, T) '.# ,$+"%/',0

1,'"2' 9 :) 5)!T !&" 3&&$4%3"%8! !" 5)! "$)" 6#67!("x3*7!" )( &;76",%5)!x2 &%e!&" #<6#67!(" (!)",! -$),

#3 #$%T2∀(a, b)∈G22xT a=xT b⇒xT(xT a) =xT(xT b)⇒(xT x)T a= (xT x)T b⇒eT a=eT b⇒a=b.

(G, T) !&" ,6+)#%!, = +3)4>! !" $( 7$(",! *! #3 7?7! @3A$( 5)! 4! +,$)-! !&" ,6+)#%!, = *,$%"!.

"#$%&'() *+ +* 34(. "( +,!"-' (G, T)5 #!"# $%$&'(# 4)&'# #!"6!",. "( "(/7"' $%$&'(# .8&$#,/7"'0 1,'"2' 9 B<!C%&"!(4! *) &;76",%5)! !&" D)&"%16! -3, #3 *61(%"%$(2 %# ,!&"! = -,$)8!, #<)(%4%"6. E)--$&$(& 5)!

∀x∈E2∃x, x′′∈E, x6=x′′2xT x=x′′T x(=e)⇔x=x′′ -)&%5)!x∈G!&" ,6+)#%!, F= *,$%"!G.

,-)&./.) 0* F(!( 9!,,/+$G H67$(",!, 5)! &% (E, ⋆) !&" )( +,$)-! *<6#67!(" (!)",! e2 *3(& #!5)!# "$)"

6#67!(" !&" /12%345/6 F4<!&"I=I*%,! x ⋆ x=eG2 3#$,& 4! +,$)-! !&" 4$77)"3"%@.

J6(6,3#!7!("2 #<)(%5)! 6#67!(" &;76",%5)! *!x !&" ($"6 x1. K( 3 7&%8&/$5$ *+ +!

• e1=e0 :*$%$&'(# ('"#,' '.# .!( -,!-,' .8&$#,/7"'5 /% '.# /(2!%"#/;0

• (x−1)−1 =x0

• (xT x)1=x′−1T x10 :' .8&$#,/7"' )*"( -,!)"/# '.# %' -,!)"/# /(2',.' )'. .8&$#,/7"'.0

• 1%". +$($,4%'&'(# (x1T x2T . . . T xn)1 =xn1T xn11T . . . T x110

0( -3,"%4)#%!, $( 3 (xn)−1 = (x−1)n2 Fn ∈NG. K( ($"! 3#$,& x−n := (xn)−1 4! 5)% -!,7!" *! *61(%, xm -$), m∈Z !( 4$(8!(3("x0 =e. H3(& 4!& 4$(*%"%$(& $( 3 #!& ,!#3"%$(&.

(xm)m =xmm !"xmT xm =xm+m2∀m, m ∈Z.

(12)

!"! #$% %&'%()*&'+$%

!"#$%$&# '()(* ! !" #$%!"&' "#$ %!" &'($)" S *+%! ,(-%&" (G, T) .%)/ 0%!)" *" 1' 1-) T *" G/ "#$

'%##) %! ,(-%&"2

+,&-&.$%$&# '()(/ ! #-%#3"!#"041"H*"(G, T)"#$ %! #-%#3,(-%&" #) "$ #"%1"0"!$ #) )1 5-!$)"!$e67170"!$

!"%$(" *" (G, T)8 "$ "#$ #$'41" &'( &(-*%)$ "$ &'##'," 9 1+)!:"(#"2

;("%:" !" #$ %&#%'($%()*+( ,( G-./"0( 1(% 2/&"% 3/&3/".2.%4 15(%2 *"($ #$ %&#%'6/&#3(7

8.1"3/&9#()($24 %"H (%2 #$ %&#%'6/&#3(4 "+ 3&%%:,( #$ .+.)($2 $(#2/(e07 !&"2 ;+&/%x #$ .+.)($2 ,(H4 &$

; xT e0 =x =xT e <+; 3/()":/( +&" (%2 3/"%( ,;$% H4 +; ,(#=":)( ,;$% G>4 ,&$1 e0 =e3;/ %")3+"01;2"&$7

?( )@)(4 +5"$-(/%( ,( x ,;$%H (%2 $.1(%%;"/()($2 #$ "$-(/%( ,;$% G.6;+()($24 ,&$1 "+ %5;6"2 ,( +5#$"9#(

"$-(/%( ,( x 3;/ T4 (2 H (%2 %2;*+( 3;/ "$-(/%"&$7 A$0$4H ,&"2 1+;"/()($2 @2/( %2;*+( 3;/ 3/&,#"27

+,&-&.$%$&# '()(' !" &'($)" H !-! :)*" *"G/ 5-!$"!'!$ e/ "#$ %! #-%#3,(-%&" *" (G, T) #) "$ #"%1"0"!$

#)

<2 ∀x, y∈H, xT y∈H

=2 ∀x∈H, x1 ∈H

>"# *"%? 5-!*)$)-!# &(757*"!$"# &"%:"!$ @$(" ("0&1'57"# &'( 1' #%):'!$" A

∀x, y∈H, xT y1∈H

0123,451 '()(/ B; $&2;2"&$ H < G (%2 ()3+&C.( 3&#/ ,"/( H (%2 %&#%'6/&#3( ,( G7 B&/%9#5&$ $5(=1+#2 3;% +; 3&%%"*"+"2. 9#( H %&"2 .6;+ DG &$ .1/"2H ≤G7

671,8$81 *) <!-! 5-((),7> E&$2/(/ 9#(

F ,;$% +( 6/&#3( )#+2"3+"1;2"G(C,×),(% $&)*/(% 1&)3+(=(%4 +5($%()*+( ,(% $&)*/(% ,( )&,#+(1(%2 #$

%&#%'6/&#3( ,( C4

F ,;$% +( 6/&#3( ;,,"2"G (F,+) ,(% G&$12"&$% $#)./"9#(%4 +5($%()*+( ,(% G&$12"&$% +"$.;"/(% x 7→ax (%2

#$ %&#%'6/&#3( ,(F7

671,8$81 ** !&"2 n ∈ N7 H$ $&2( nZ +5($%()*+( ,(% ($2"(/% /(+;2"G% ,( +; G&/)( nx4 x ∈ Z7 E&$2/(/

9#( nZ(%2 #$ %&#%'6/&#3( ;,,"2"G ,( Z7

B-(("5$)-! !&"2 nZ={nk, k∈Z}={. . . ,−2n, n,0, n,2n, . . .}.

F !&"2 (x1, x2) ∈ (nZ)24 ∃(k1, k2) ∈ Z24 x1 = nk14 x2 = nk2 ⇒ x1+x2 = n(k1+k2) ⇒ x1 +x2 ∈ nZ ,&$1 +; +&" (%2 "$2(/$(7

<' !&"2 (x1, x2, x3) ∈ (nZ)34 ∃(k1, k2, k3) ∈ Z34 x1 = nk14 x2 = nk24 x3 = nk37 H/ (x1 +x2) +x3 = (nk1+nk2) +nk3 =nk1+ (nk2+nk3) =x1+ (x2+x3) ,&$1 +; +&" (%2 ;%%&1";2"-(7>

<' 0Z= 0nZ = 07 A$ (I(24 %"x∈nZ4x+ 0 = 0 +x=x7>

F !&"2 x∈nZ4∃k∈Z4x=nk7∃x=n(−k)∈nZ4x×x =n(k−k) = 0 ,&$1 2&#2 .+.)($2 3&%%:,( #$

%C).2/"9#(7

J&$1+#%"&$4 nZ(%2 #$ %&#%'6/&#3( ;,,"2"G ,( Z7

671,8$81 *9 E&$2/(/ 9#( +5($%()*+( ,(% /;1"$(%n'":)(% ,( +5#$"2. G&/)($2 #$ %&#%'6/&#3( ,((U,×)7 B-(("5$)-! H$ 1&$%2;2( ,;$% #$ 3/()"(/ 2()3% 9#( +5($%()*+( Un ,(% /;1"$(% n'":)(% ,( +5#$"2. (%2 #$

%&#%'($%()*+( ,(U7 A$ (I(24 %"zn= 14 &$ ; ($ 3;/2"1#+"(/|z|n= 14 ,&$1|z|= 1<,;$%R+4 +5.9#;2"&$xn= 1

; #$( %(#+( %&+#2"&$>7 A$%#"2(4 "+ /(%2( D G;"/( +(% -./"01;2"&$% .+.)($2;"/(% Un 1&$2"($214Un (%2 %2;*+( 3;/

(13)

!"#$%& '(% zn =zn = 1) *+"!( (zz)n = znzn = 1, -& *! %./-!(- '(% zn = 1) 1

zn = 1,) #".0 01-(& 2%-. $.

("$(34!"$ - 5$+&% +%0*&%6 #-U7

!"#$%&%# '( 8. 0".(%#9!- +1-.(-52+- 0".(&%&$: #-( (%; 6".0&%".( #-R− {0,1} #*.( +$%35<5- ($%/*.&-( = f1(x) =x) f2(x) = 1

1−x) f3(x) = x

x−1) f4(x) = 1

x) f5(x) = 1−x) f6(x) = x−1 x 7

>".&!-! ?$1%+ (1*4%& #1$. 4!"$ - "$! +* 0"5 "(%&%". ':0!%!- (* &*2+-,7 @:&-!5%.-! &"$( (-( ("$(34!"$ -(7 !""#$%&!' = A- +$( (%5 +- -(& #- 6*%!- $. &*2+-*$ #- +"% =

" f1 f2 f3 f4 f5 f6 f1 f1 f2 f3 f4 f5 f6 f2 f2 f6 f4 f5 f3 f1 f3 f3 f5 f1 f6 f2 f4 f4 f4 f3 f2 f1 f6 f5 f5 f5 f4 f6 f2 f1 f3

f6 f6 f1 f5 f3 f4 f2

B"$! "2&-.%! &"$( +-( ("$(34!"$ -() +- +$( (%5 +- -(& #- +-( 0".(&!$%!- -&%& C -&%&7 8. 0"..*D& +-( ("$(3 4!"$ -( &!%/%*$; = +- 4!"$ - G &"$& -.&%-! -& +- ("$(34!"$ - !:#$%& C +1:+:5-.& .-$&!-7 B*! *%++-$!() &"$&

("$(34!"$ - 0".&%-.& f1 ?$% -(& +- .-$&!-7 E% ". 0F-!0F- $. ("$(34!"$ - 0".&-.*.&f1 -&f2) ". /"%& ?$- "$!

<&!- (&*2+- *! G"H %+ #"%& *$((% 0".&-.%! f2of2=f67 8. 0".(&*&- ?$-{f1, f2, f6}-(& $. &!"%(%95- ("$(34!"$ -

#- G7 B*! 0".&!- (% ". *I"$&-f3) f4 "$ f5 C f1 -& f2) ". -(& "2+%4: "$! */"%! (&*2%+%&: #1*I"$&-! &"$&-( +-(

*$&!-( 6".0&%".(7 J-5*!?$".( -.($%&- ?$- {f1, f3} -(& $. ("$(34!"$ - #- G) 5*%( ?$- (% ". K *I"$&- $.-

&!"%(%95- 6".0&%".) ". /* C ."$/-*$ !-&"52-! ($! +- ("$(34!"$ - &!%/%*+G7 @- 5<5-){f1, f4}-&{f1, f5}(".&

#-( ("$(34!"$ -() -& ". .1-. "2&%-.& *( #1*$&!-(7 A- 4!"$ - G * #".0 $. ("$(34!"$ - C $. :+:5-.&) &!"%(

("$(34!"$ -( C #-$; :+:5-.&( -& $. C &!"%( :+:5-.&() -& +$%35<5- -(& $. ("$(34!"$ - C (%; :+:5-.&(7

)$*+*,&-&*. /0 0 (& (G, T) #)% *' +"!*,#- )& H #% K )!'% .#*/ )!*)0+"!*,#) .# G- 12!") H ∪K #)% *' )!*)0+"!*,# .# G )& #% )#*2#3#'% )& H⊆K !*K ⊆H4

5"#*6# = E"%-.& H -& K #-$; ("$(34!"$ -( #- G &-+( ?$- H∪K ("%& $. ("$(34!"$ - #- G7 L"$( #-/".(

!"$/-! ?$1$. #-( #-$; ("$(34!"$ -(H"$K -(& 0".&-.$ #*.( +1*$&!-7 E$ "(".( ?$-H.- ("%& *( 0".&-.$

#*.(K-& !"$/".( ?$-K-(& 0".&-.$ #*.(H7 E"%&k$. :+:5-.& #-K7 M+ (1*4%& #- !"$/-! ?$-k* *!&%-.&

CH7 B$%(?$- H.1-(& *( 0".&-.$ #*.(K) %+ -;%(&-h∈H &-+ ?$-h /∈K7 B$%(?$-H∪K -(& $. ("$(34!"$ -

#-G) ."$( */".(hT k∈H∪K7 @1*$&!- *!&) $%(?$-h /∈K -&k∈K) ."$( */".(hT k /∈K) #".0hT k∈H)

#1"N k∈H 0- ?$% !"$/- ?$-K ⊆H7

A* !:0% !"?$- -(& :/%#-.&- = (% *! -;-5 +-)H ⊆K)H∪K=K-&H∪K-(& .*&$!-++-5-.& $. ("$(34!"$ -7

1#23$45# /0 0/ H∪K.1-(& *( -. 4:.:!*+ $. ("$(34!"$ - #-G7 B*! -;-5 +-) ("%-.&H+* #!"%&- #1:?$*&%".

y = 0-&K +* #!"%&- #1:?$*&%".x= 0#*.(R2) 4!"$ - *##%&%67 O+"!(H-&K (".& #-( ("$(34!"$ -( #-(R2,+) 5*%( *(H∪K 0*!(1,0) + (0,1) = (1,1).1* *!&%-.& *( C H∪K7

)$*+*,&-&*. /0 0' (!&#'%(G, T)*' +"!*,# #%F *'# 713&22# '!' 6&.# .# )!*)0+"!*,#) .#G8F ,#*% $!'%#'&"

*' '!39"# :'& !* &':'& .# )!*)0+"!*,#);4 (& I #)% 2<&'%#")#$%&!' .# %!*) 2#) =2=3#'%) .# F- 1*%"#3#'% .&%

I =∩HFH 12!") I #)% 2*&03>3# *' )!*)0+"!*,# .# G4

(14)

!"#$" !"#$%&'("#$ $)#$ *+(,+ &+ -'#'().%,' ./%#,+($+0,%"# I =H∩K &+ &+12 $"1$3-("1*+$H +,K4H∩K

#/+$, *)$ 5%&+ 0)( e )**)(,%+#, 6 H ∩K4 7"%+#, x +, y )**)(,+#)#, 6 H ∩K4 !"88+ H +, K $"#, &+$

$"1$3-("1*+$ &+ G9xT y−1 )**)(,%+#, 6 H +, 6K &"#0 6H∩K4 :/";9 H∩K +$, 1# $"1$3-("1*+ &+G4 <)

&'8"#$,(),%"# $+ -'#'().%$+ 6 1# #"8=(+ >#% "1 %#>#% &+ $"1$3-("1*+$4

"#$%&'&(% )* *+ %&'( (G, T) #) *!&#+" "( A #) ,&#,-"),"./0" )&) $'1" 1" 23 4) 5++"00" !" #$%!"&' '($'()%* +5! A 0" ,&#,-*!&#+" < A >= ∩HSH &6 S ",( 07"),"./0" 1" (&#, 0", ,&#,-*!&#+", 1" G 8#' 9&)('"))")( A3 %'g 5++5!('")( :G; &) )&(" < g > : 05 +059" 1"G3

!+, +#$+8=.+ #/+$, *)$ 5%&+ 0)( %. 0"#,%+#,A.1%38?8+ +, .) @"(81.+ 0%3&+$$1$ +$, *)( 0"#$'A1+#, =%+# &'>#%+4 ,-./01. )* *) < m >=mZ9< G >=G9< m, n >=pgcd(m, n)Z$"#, &+$ $"1$3-("1*+$ +#-+#&('$4 23#(45/. )* * <" ,&#,-*!&#+"< A >",( 0" +0#, +"('( ,&#,-*!&#+" =5# ,"), 1" 07')90#,'&)> 1"G9&)(")5)(

A3 ?#(!".")( 1'(;< A > ",( 95!59(@!',@ +5! 0", 1"#A 9&)1'('&), ,#'$5)(", B C3 < A > ",( #) ,&#,-*!&#+" 1" G9&)(")5)( A3

D3 %'K ",( #) 5#(!" ,&#,-*!&#+" 1" G9&)(")5)( A &) 5 < A >⊂K3

!"#$" 7"%, < A >.+ $"1$3-("1*+ +#-+#&(' *)(A4< A >=∩H"; ./%#,+($+0,%"# *"(,+ $1( .+$ $"1$3-("1*+$

H &+ G 0"#,+#)#,A4< A > +$, &"#0 1# $"1$3-("1*+ &+ G 0"#,+#)#,A &/)*(B$ .) *("*"$%,%"# C4D4E4 7"%, K 1# $"1$3-("1*+ &+ G 0"#,+#)#, A4 F."($9 K @)%, *)(,%+ &+ ./+#$+8=.+ &+$ $"1$3-("1*+$ $1( .+A1+. *"(,+

./%#,+($+0,%"# &'>#%$$)#, < A > &"#09< A > ',)#, %#0.1$ &)#$ ,"1, $"1$3-("1*+ &+ G0"#,+#)#,A9< A >

+$, %#0.1$ &)#$ K4 G)( 0"#$'A1+#,9< A >+$, .+ *.1$ *+,%, $"1$3-("1*+ &+ G0"#,+#)#,A4

64(0(7&'&(% )* *+ E&#( ,&#,-*!&#+" 1" Z ",( 1" 05 F&!." < n >=nZ &6 n",( #) ")('"! +&,'('F3 !"#$" 7"%, H 1# $"1$3-("1*+ &+Z&%$,%#0, &+ Z4

H 7%H = 0)."($H =<0>4 H 7%H =Z9H=<1>4

H I# $1**"$+ A1+H +$, 1# 7(879:4(80. 04(04. &+ Z J&"#0 &%K'(+#, &+ Z +, &+{1}L4 7"%, n .+ *.1$

*+,%, '.'8+#, $,(%0,+8+#, *"$%,%@ &+H9 8"#,("#$ )."($ A1+< n >=H4

n)**)(,%+#, 6 H $"1$3-("1*+ &+Z &"#0< n > +$, %#0.1$ &)#$H J< n >*.1$ *+,%, $"1$3-("1*+ &+Z 0"#,+#)#,nL4

M"#,("#$ A1+ H +$, %#0.1$ &)#$ < n > $"%, x )**)(,+#)#, 6 H4 :/)*(B$ .) &%5%$%"# +10.%&%+##+9 %.

+2%$,+ 1# 0"1*.+ &/+#,%+($ (i, j) )5+0 0≤j < n ,+. A1+x=in+j4

7%i+$, *"$%,%@9 in=n+...+n)**)(,%+#, 6H 0)(n)**)(,%+#, 6 H +,H +$, 1# $"1$3-("1*+ &+ Z4 7%i+$, #'-),%@9 in= (−n) +...+ (−n) )**)(,%+#, 6 H 0)(n )**)(,%+#, 6 H +,H +$, 1# $"1$3-("1*+

&+ Z4

!"88+ x)**)(,%+#, 6H $"1$3-("1*+ &+ Z9j =x−in )**)(,%+#, 6H4 I(j +$, *"$%,%@ +, $,(%0,+8+#,

%#@'(%+1( 6 n+,n+$, .+ *.1$ *+,%, +#,%+( $,(%0,+8+#, *"$%,%@ )**)(,+#)#, 6H &"#0 j= 04 :/";9 x=in +,x )**)(,%+#, &"#0 6< n >4H +$, %#0.1$ &)#$ < n >4 !"#0.1$%"#9 H=< n >=nZ4

,-.4;&;. !< J)&) 9&!!'*@L

N4 7"%+#, mZ+,nZ&+12 $"1$3-("1*+$ &+ Z4 M"#,(+( A1+

mZ+nZ={mu+nv|u, v∈Z} )L +$, 1# $"1$3-("1*+ &+Z9

(15)

! "#$%&'$% mZ'%nZ(

"! ')% "#$%'$* +,$) %#*% )#*)-./#*0' +'Z 1*& "#$%&'$% mZ'%nZ2 +! 3&mZ+nZ=dZ( 1*' 0'*%-#$ +&/' +' d4

52 67%'/8&$'/ 9') )#*)-./#*0') '$.'$+/7) 0,/ : 14Z∪35Z ; 4Z∪8Z∪6Z∪64Z ; 2Z∪3Z ; 4Z∪21Z ; 5Z∪25Z∪7Z ; {70,4}2

<$ &$%/#+*&% '$)*&%' 9, $#%&#$ +' )#*)-./#*0' +&)%&$.*7 =*%&9&)7' &$&%&,9'8'$% 0,/ >,9#&)!2 ?') )#*)-./#*0') +&)%&$.*7) "#$$,&))'$% +') ,009&",%&#$) '$ .7#87%/&' +,$) 9@7%*+' +') ,"%&#$) +' ./#*0')( '$ %#0#9#.&' ,9.7- /&1*' +,$) 9, "9,))&A",%&#$ +') /'BC%'8'$%)( '$ %D7#/&' +' >,9#&) +,$) 9, "#//')0#$+,$"' +' >,9#&)2

"#$%&'&(% )* *+ ! "#$"%&'#$() H *+$! &'#$() (G, T) )", !"#!$%&' -#$ $()*+, #$ !$-+)!+$#. "/ (#$' ,#$, x *) G ), (#$' ,#$, h *) H0 1) ('#*$/, xT hT x1 )", 2123)!, *) H4 5! !#,) 61#'" H ⊳ G ), #! 1/, 7H )", */",/!&$2 *6!" G84

E$ )#*)-./#*0' +&)%&$.*7 H +@*$ ./#*0' G ')% *$ )#*)-./#*0' .9# ,9'8'$% )%, 9' 0,/ 9@,"%&#$ +' G )*/

9*&-8C8' 0,/ "#$F*.,&)#$2 ?') )#*)-./#*0') +&)%&$.*7) &$%'/B&'$$'$% $,%*/'99'8'$% +,$) 9, +7A$&%&#$ +*

1*#%&'$% +@*$ ./#*0'2

"#$%&'&(% )* *, 9#$' ,#$, x *)G #! !#,) :

; xH 1+)!")3<1) -.,+""/ 0 %+&.1/. *)" 2123)!," *) G *) 16 =#'3)xT h 6>)? h 2123)!, *)H0

; Hx 1+)!")3<1) -.,+""/ 0 )(!#/. *)" 2123)!," *) G *) 16 =#'3)hT x 6>)?h 2123)!, *)H0

; xHx1 1+)!")3<1) *)" 2123)!," *)G *) 16 =#'3)xT hT x1 6>)? h 2123)!, *)H4

E$' G,H#$ 71*&B,9'$%' +' +7A$&/ *$ )#*)-./#*0' +&)%&$.*7 ')% +' +&/' 1*' 9') "9,))') I +/#&%' '% I .,*"D' +' H +,$)G"#J$"&+'$%( "@')%-I-+&/' :

-.(/(0&'&(% )* *+

H ⊳ G "/ ), ")$1)3)!, "/ xHx1 =H⇔xH =Hx

9')$>) : K,/ +7A$&%&#$( H ⊳ G ⇔ xHx1 ⊂H 0#*/ %#*% x +' G2 </( H ⊂ xHx12 L$ 'M'%( )& h ∈ H( #$

,*/, :

h∈xHx1⇔ ∃h∈H(h=xT hT x1⇔ ∃h ∈H(h =x1T hT x

N* 1*' H ⊳ G( h = x1T hT x ')% 'M'"%&B'8'$% 7978'$% +' H2 O&$)&( #$ 0'*% 7.,9'8'$% +&/' H ⊳ G )& '%

)'*9'8'$% )& xHx1=H⇔xH =Hx2

1234/53 )* * ?') +709,"'8'$%) +* 09,$( "#$)%&%*7) +') %/,$)9,%&#$) '% +') /#%,%&#$)( "#$)%&%*'$% *$ ./#*0'

=$#$ "#88*%,%&G! 0#*/ 9, 9#& +' "#80#)&%&#$ +') ,009&",%&#$)2 ?') %/,$)9,%&#$) '$ "#$)%&%*'$% *$ )#*)-./#*0' +&)%&$.*72

6347.893 )* *

P {e} '%G)#$% %#*F#*/) +') )#*)-./#*0') +&)%&$.*7)2

P E$ )#*)-./#*0' +'G +' 9, G#/8'xHx1 ,B'" x∈G')% ,00'97 *$ :(%;9<9# +' H2 K,/ "#$)71*'$%(

*$ )#*)-./#*0' ')% +&)%&$.*7 )& '% )'*9'8'$% )@&9 ')% )#$ )'*9 "#$F*.*72

123.:&:3 => 3#&% (G,×) *$ ./#*0' '% ≃ *$' /'9,%&#$ +@71*&B,9'$"' )*/ G2 <$ )*00#)' 1*' "'%%' /'9,-

%&#$ ')% "#80,%& 9' ,B'" 9, 9#& +' ./#*0'( "@')%-I-+&/' 1*' )& ∀x, y, x, y∈G(x≃x '%y≃y ,9#/)xy≃xy2

(16)

!"#$%$ &'% () *()++% H ,% (-.(./%"# "%'#$%1 %+# '" +!'+01$!'2% ,3+#3"1'. ,% G %# &'% ∀x, x ∈G4x ≃x

%+# .&'35)(%"# 6 xx1 ∈H7

!""#$%&!' 8 9#)"# ,!"".+y, z ∈H4 !" )y≃1 %#z≃17 :) *!/2)#3;3(3#. ,% () (!3 ,!""% ,-'"% 2)$#yz≃14 +!3# yz ∈H4 %# ,-)'#$% 2)$# yy1 ≃y1 +!3# y1 ∈H7 <%() /!"#$% &'% H %+# '" +!'+01$!'2% ,% G7 =!'$

#!'# x ∈ G4 !" ) )'++3 xyx1 ≃ x1x1 = 1 %# ,!"* xyx1 ∈ H7 :% +!'+01$!'2% H %+# ,!"* ,3+#3"1'.7

>% 2('+4 2!'$ x, x ∈ G4 +3 x ≃ x4 )(!$+ 2)$ *!/2)#3;3(3#. ,% () (!34 !" ) xx−1 ≃ xx−1 = 14 *-%+#060,3$%

xx1 ∈H7

?.*32$!&'%/%"#4 +3 xx1 ∈H4 )(!$+ xx1≃14 %# ,!"*4 2)$ *!/2)#3;3(3#. ,% () (!34x≃x7

"#$%$&'('$) *+ +, (&G #)% $!**+%,%&- ./"!+0# ,123&#'45 ,3!") %!+% )!+)6/"!+0# 7#G #)% 7&)%&'/+2 7,')G8 9"#+:# 8 @!3# H '" +!'+01$!'2% ,%G7(G, T) %+# *!//'#)#3A ,!"* #!'+ (%+ .(./%"#+ ,%G*!//'#%"# %"#$%0

%'B C *!/2$3+ *%'B &'3 +!"# ,)"+ H D(H, T) %+# ,!"* *!//'#)#3AE7 =)$ *!"+.&'%"#4 ∀x ∈ G %# ∀h ∈ H4 xT h =hT x⇔ xT hT x1 =h %" *!/2!+)"# 6 ,$!3#% 2)$ x1 D(-3"5%$+% ,% x4 &'3 %B3+#% 2'3+&'% (G, T) %+#

'" 1$!'2%E7 <!//% h∈H4 !" *!"*('# &'% #!'# +!'+01$!'2% H ,% G%+# ,3+#3"1'. ,)"+G7

-./0%1/ *+ +2 (Z,+) %+# *!//'#)#3A %#nZ%+# '" +!'+01$!'2% ),,3#3A ,% Z,!"* nZ⊳Z7 !"!" #$%&'()*'+$% ,) -)$'+.%' ,/)% 0($)1.

:%+ +!'+01$!'2%+ ,3+#3"1'.+ +!"# 3/2!$#)"#+ ,)"+ (-.#',% ,%+ 1$!'2%+ &'!#3%"#+ 6 *)'+% ,' $.+'(#)# +'30 5)"# 8 !" 2%'# *!"+#$'3$% '" 1$!'2% &'!#3%"#G/H ,% (!3 *!/2)#3;(% )5%* *%((% ,%G+3 %# +%'(%/%"# +3H %+#

'" +!'+01$!'2% ,3+#3"1'. ,% G7 =('+ 2$.*3+./%"#4 ,)"+ (-.#',% ,%+ 1$!'2%+4 (% &'!#3%"# ,-'" 1$!'2% %+# '"%

!2.$)#3!" *()++3&'% 2%$/%##)"# () *!"+#$'*#3!" ,% "!'5%)'B 1$!'2%+ 6 2)$#3$ ,-)"*3%"+7 F 2)$#3$ ,-'" 1$!'2%

G4 %# ,-'" +!'+01$!'2% H4 !" 2%'# ,.G"3$ '"% (!3 ,% 1$!'2% +'$ (-%"+%/;(%G/H ,%+ *()++%+ ,% G+'35)"#H4 6 *!",3#3!" &'% (%+ *()++%+ ()#.$)(%+ ,$!3#%+ +!3%"# .1)(%+ )'B *()++%+ ()#.$)(%+ 1)'*H%+ DxH =HxE7

@!3%"# (G, T) %#(H, T) ,.+31")"# $%+2%*#35%/%"# '" 1$!'2% %# '" +!'+ 1$!'2% ,% G7

"#$%$&'('$) *+ +3 (!&#'% x, y∈G8 ;, "#3,%&!' R 72<'&# 0," xRy ⇔ xT y1 ∈ H #)% +'# "#3,%&!' 7=2>+&6 :,3#'$#.? 7"!&%#4 )+" G8

9"#+:# 8

I R%+# $.J%B35% 8∀x∈G4xT x1 =e∈H⇔xRx7

I K"+'3#%4 *!//% H %+# '" +!'+01$!'2% ,% G4 #!'# .(./%"# ,)"+ H ),/%# '" 3"5%$+% D,)"+ HE ,!"*

(xT y1)1 =yT x1 ∈H ⇔yRx DH .#)"# +#);(% 2)$ 3"5%$+3!"E7 =)$ *!"+.&'%"#4 R%+# +C/.#$3&'%7 I !"#$!"+ %"G" &'%R%+# #$)"+3#35% 8 +!3#(x, y, z)∈G34(xRy %#yRz)⇔(xT y1 ∈H %#yT z1 ∈H)7

(xT y1)T(yT z1) =xT(y1T y)T z1 =xT eT z1 =xT z1 ∈H⇔xRz D*)$ H %+# +#);(% 2!'$ () (!3 ,% *!/2!+3#3!"E7

=)$ *!"+.&'%"#4 D∀x, y∈H4xRy)⇔(H %+# '" +!'+01$!'2% ,% GE ,-)2$L+ () 2$!2!+3#3!" M7N7M 7

"#$%$&'('$) *+ +4 (!&% x ∈ G8 ;, $3,))# 7=2>+&:,3#'$# .? 7"!&%#4 7# x 0!+" 3, "#3,%&!' R #)% 3=#')#*13#

Hx={hT x, h∈H}8 @33# #)% '!%2# cl(x) !+ x8

9"#+:# 8 =)$ ,.G"3#3!"4 cl(x) = {y∈G, xRy}7 @!3# y ∈G .&'35)(%"# 6x 2!'$ () $%()#3!" R7 O(!$+ 3( %B3+#%

k ∈H D,!"# (-3"5%$+% k1 =h %+# )'++3 ,)"+HE #%( &'% xT y1 =k⇔ y =hT x7 K# ,!"* y %+# .(./%"# ,%

Hx7 ?.*32$!&'%/%"#4 +3y %+# .(./%"# ,% Hx4 3( %+# *()3$ &'%xRy7

5/06#78/ *+ +2 P" )'$)3# )'++3 2' ,.G"3$ () $%()#3!" ,-.&'35)(%"*% D6 1)'*H%ER2)$xRy⇔y1T x∈H7

>)"+ *% *)+4 () *()++% ,-.&'35)(%"*% ,-'" .(./%"#x,%G)'$)3# .#. ,!"".% 2)$ (-%"+%/;(%xHD*()++% 6 1)'*H%

,% x ,% HE7

(17)

!""#$#%$ &!' H $#() !% $#!$*+,#!"' -($)(%+!. -' G/ 0% -.$(,' -.1%(, !%' 2#( 3 $!, 24'%$'562' &!#)('%) G/R ! "# $#%&' "# (")* '#+),!""! (&**-."! /0!*+121 -,! !' (&*#'+cl(x) 3cl(y) =cl(xT y)4 5# 6)*+-7/#+-&' ! /! /8&-9 !*+ "# *)-:#'+! ; *- X !+Y *&'+ !)9 /"#**!* 0<"<=!'+* ! G *)-:#'+ H>XY !' !*+ )'! #)**-4 ?"

!9-*+! !* <"<=!'+* x !+y ! G +!"* @)!X !+Y *&-!'+ ,!(!/+-:!=!'+ "!* /"#**!* ! x !+y *)-:#'+H4 A&)*

#:&'* #"&,* XY = (xH)(yH) = (Hx)(yH) = H(xT y)HB /&==! H !*+ -*+-'C)<> '&)* (&):&'* ,!=("#/!, H(xT y) (#, (xT y)H !+ '&)* +,&):&'* XY = xT yHH4 D#-* HH =H E()-*@)! H !*+ )' *&)*1C,&)(! ! GF> &'/ "# ,!"#+-&' &.+!')! (!)+ *0</,-,! XY =xT yH> /! @)- =&'+,! .-!' @)!XY !*+ )'! /"#**! *)-:#'+

H E!+ (")* (#,+-/)"-G,!=!'+> "# /"#**! !xT yF4 H! /! @)- (,</G !> -" ,<*)"+! @)0!' $#-*#'+ /&,,!*(&' ,! 2 )'!

/"#**! X !+ )'! /"#**! Y "0!'*!=."! XY> '&)* <7'-**&'* )'! "&- ! /&=(&*-+-&' 3 #'* "0!'*!=."! !*

/"#**!* *)-:#'+H !+ @)! /!++! "&- (!)+ I+,! /#,#/+<,-*<! (#, "# ,!"#+-&'

xH 3yH = (xT y)H ⇔cl(x)3cl(y) =cl(xT y)4

J!++! <7'-+-&' # )' *!'* *- !+ *!)"!=!'+ *- "# /"#**! cl(xT y) '! <(!' (#* ) /8&-9 !* ,!(,<*!'+#'+*x !+

y !* /"#**!* cl(x) !+cl(y)> #)+,!=!'+ -+ *- ; (&),x, y, x, y∈G>

xRx yRy

⇒xT yRxT y. EK4LF

M- "0-=("-/#+-&' EK4LF !*+ :<,-7<!> &' -+ @)! "# ,!"#+-&' 0<@)-:#"!'/! R !*+ /&=(#+-."! #:!/ "# "&- T E/$4 /8#(-+,! LF4 N' #6&)+! "# <7'-+-&' *)-:#'+! ;

!"#$%$&# '()(* ! "#$!%&'( R #)% *&%# !"#$%&'() * +,!&%) +"#),#-%&.#/#(% * -$. /)0 !.#- $! $'&T )&

∀x, y, z ∈G1 xRy⇒xT zRyT z +"#),#-%&.#/#(% xRy⇒zT xRzT y0 N' # #"&,* "!

+,!&-./0 '()(1 ! "#$!%&'( R #)% -'/,!%&2$# !.#- $! $'& *# 3"'4,# )& #% )#4$#/#(% )& #$$# #)% -'/,!%&2$# 5

*"'&%# #% 5 3!4-6#7

8"#4.# ; 5! *!'* '&' +,-:-#" *! =&'+,! #-'*- ; xRx !+yRy ⇒xT yRxT y !+xT yRxT y⇒xT yRxT y4 N' /&'*- G,! !'7' "! +8<&,G=! -'+!,=< #-,! *)-:#'+ ;

+,!&-./0 '()(2 9$ : ! ;<4&.!$#(-# #(%"#

=7 ! "#$!%&'( *>;<4&.!$#(-# R#)% -'/,!%&2$# 5 *"'&%#7

?7 9$ #@&)%# 4( )'4)A3"'4,# H %#$ <4# xRy⇔xT y1∈H⇔y∈Hx

8"#4.# ; M- R !*+ /&=(#+-."! 2 ,&-+!> xRy ⇔ xT y1Re⇔ xT y1 ∈cl(e) !+ &' :<,-7! @)! H = cl(e) !*+

.-!' )' *&)*1C,&)(! !G4 O</-(,&@)!=!'+ *- &' #2.>xT y1∈H⇔(xT z)T(yT z)1 ∈H>R!*+ /&=(#+-."!

2 ,&-+!4

H!* !)9 +8<&,G=!* (,</< !'+* &' < )-+ ;

+,!&-./0 '()(3 B'&% (G, T) 4( 3"'4,# #%R 4(# "#$!%&'( *>;<4&.!$#(-# *!() G7 9$ : ! ;<4&.!$#(-# #(%"# C

=7 cl(x) Dcl(y) =cl(xT y) *;E(&% 4(# $'& *# 3"'4,# )4" G/R7

?7 ! "#$!%&'(R #)% -'/,!%&2$# 5 *"'&%# #% 5 3!4-6# !.#- $! $'& *# 3"'4,#7 F7 9$ #@&)%# 4( )'4)A3"'4,# H %#$ <4#

+!0 ∀x∈G1xH =Hx⇔H #)% *&)%&(34; *!() G7 +20 xRy⇔xT y1 ∈H ⇔x1T y∈H7

(18)

!"# $!%$&"#' !% (!%%' &)

"#$%&'&(% )* *+ !"# H ⊳ G$ %& '()&#"!* R+,-*"( .&'

xRy⇔xT y1 ∈H

(/# &..(),( !"#$%&' ()*+,%-#"!'.! /0"1&*# )( /!0/23'!0.( H$ %4(*/(56)( 70!#"(*# G/R(/# &)!'/ 0* 3'!0.(

.!0' )& )!" *&#0'())(8 &..(), / &,0! +,&$%!'$8 (# /( *!#( G/H$

,-./01. )* *2 *!+, &) #'&),+!% R (-.%+' /"# Z 0)# xRy ⇔ ∃k ∈ Z/x−y = kn1 R '/, "%' #'&),+!%

(2-3"+4)&'%$'5 6',,' #'&),+!% '/, )00'&-' 3.14'&(% 5. 6(%738.%6. /(581(n5 7'"8 '%,+'#/x',y'% #'&),+!%

/!%, (+,/ $!%9#"/ &2"% : &2)",#' ;!("&! n ', !% %!,' )&!#/x ≡y ;!( n <!" x ≡y[n]=5 >% #';)#3"'#) 3"' x≡0;!( n/+ ', /'"&';'%, /+n(+4+/' x5

*!+, x )00)#,'%)%, : Z5 >% %!,' 0)# x &) $&)//' (2-3"+4)&'%$' (' x 0!"# &) #'&),+!% (' $!%9#"'%$' ;!("&!

n5 >% %!,' 0)# Z/nZ&2'%/';?&' ('/ $&)//'/ (2-3"+4)&'%$'/ (' &) #'&),+!% (' $!%9#"'%$' ;!("&!n5 >% 0'", )&!#/ (-("+#' ('/ #-/"&,),/ 0#-$-('%,/ 3"' Z/nZ'/, "% 9#!"0' <)?-&+'%= 0!"# &) &!+ x+y=x+y5

*+ !% $!%/+(@#'Z/5Z={0,1,2,3,4}1 $'&) %!"/ 0'#;', (2)A#;'# 0)# '8';&' 3"'9 = 4 + 5 = 4 + 5 = 4 + 0 = 4 + 0 = 4.

,-.36&6. 9+ >% (-.%+, /"# R2 &) #'&),+!% ρ 0)#

(x, y)ρ(x, y)⇔x+y=x+y B5 C!%,#'# 3"' ρ '/, "%' #'&),+!% (2-3"+4)&'%$'5

D5 E#!"4'# &) $&)//' (2-3"+4)&'%$' (" $!"0&' (0,0)5 F5 *!+, f &2)00&+$),+!% (' R2 ()%/R(-.%+' 0)# G

f : (x, y)→x+y

C!%,#'# 3"' ('"8 -&-;'%,/ (' R2 -3"+4)&'%,/ ;!("&! ρ !%, ;H;' +;)9' 0)# f ', 3"' ('"8 -&-;'%,/

%!% -3"+4)&'%,/ !%, ('/ +;)9'/ (+/,+%$,'/5

I5 J% (-("+#' 3"2'%,#' &2'%/';?&' 3"!,+'%, R2/ρ',R +& '8+/,' "%' ?+K'$,+!%g 3"' &2!% 0#-$+/'#)5 9!''(:#"!* G

B5 >% ;!%,#' )+/-;'%, 3"'ρ '/, #-L'8+4'1 /M;-,#+3"' ', ,#)%/+,+4' (!%$ρ '/, ?+'% "%' #'&),+!% (2-3"+4)N

&'%$'5

D5 O) $&)//' (2-3"+4)&'%$' (" $!"0&' (0,0)'/, $!%/,+,"-' 0)# &2'%/';?&' ('/ $!"0&'/ (x, y) ∈R2 4-#+.)%, x+y= 05 62'/, &2'%/';?&' ('/ 0!+%,/ /+,"-/ /"# &) ('"8+@;' ?+//'$,#+$' (" 0&)% x0y5

F5 *!+'%,u, v∈R21 )&!#/uρv⇔f(u) =f(v)5 >% '% (-("+, -9)&';'%, 3"' ('"8 -&-;'%,/ %!% -3"+4)&'%,/

!%, ('/ +;)9'/ (+/,+%$,'/5

I5 g '/, &2)00&+$),+!% 3"+ : "%' $&)//' P)+, $!##'/0!%(#' &) /!;;' ('/ $!;0!/)%,/ (2"% #'0#-/'%,)%, 3"'&$!%3"' (' $',,' $&)//'5g'/, +%K'$,+4' (2)0#@/ &) 3"'/,+!% 0#-$-('%,'5 )# )+&&'"#/1∀α∈R1 !% 0'",

$!%/+(-#'# 3"'α '/, &2+;)9' (' (α2,α2)∈R2 (!%$ g '/, /"#K'$,+4' ', 0)# $!%/-3"'%, ?+K'$,+4'5 !"!# $%&%&%'()*+&,+ -, .'%/(,+

>% (!+, : C)#+' J%%';!%( 6);+&&' Q>R7ST <BUFUNBVDD= &' $!%$'0, (2:(/(/(30:&;/. '%,#' ('"8 /,#"$,"#'/ G

"#$%&'&(% )* *<= ;* 1& 02%31! (! / &,0!3 !0 2&1&1& 02%31! (! / &,0!3 <+0 3'(: =!5!"!/ >

/(56)&6)( (# 5!'.=? > @!'5(A (/# 0*( &..)":&#"!* (*#'( +(0B 3'!0.(/ 70" '(/.(:#( )& /#'0:#0'( +(/ 3'!0.(/$

(19)

!"# $%&'(#&)*+,- #( (G, T) *,(G, ⋆) #.+, /*"0 1%."$*# /2&!&)*+,# +*",%*# %*#$*',(3#e*,n- "+* 4$$!('4,(.+

f :G→G *#, "+ ).%$5(#)* /* 1%."$*# !.%#6"* 7

∀(a, b)∈G×G-f(aT b) =f(a)⋆ f(b)

8+ ).%$5(#)* /* 1%."$*# ,%4+#$.%,* !4 !.( /* 1%."$*- *, 94 4(+#( '.+#*%9*% ,.",*# !*# $%.$%(&,&# !(&*# : '*,,*

!.(; <! *#, /.+' (+,&%*##4+, /2&,"/(*% '.))*+, #* '.)$.%,*+, !*# $%(+'($4"0 .=>*,# /* !4 ,5&.%(* /*# 1%."$*#

$4% !*# ).%$5(#)*#;

"#$%&'&(% )* *++ ! !"#"$%& !#' f "#$ %! &'('(')*&+#(" ,+-".$+/0

,-(.(/&'&(% )* *+0 1+ (G, T) "$ (G, ⋆) #'!$ 2"%3 4)'%*"# 5%"6.'!5%"#7 "$ #+ f "#$ %! +#'(')*&+#(" 2" G

#%) G7 86')# f1 "#$ %! +#'(')*&+#(" 2"G #%) G0

9)"%:" 7 ?( f *#, "+ (#.).%$5(#)*- 4!.%#f *#, "+* =(>*',(.+- /.+'f1 4"##(; <! #"@, /* ).+,%*% 6"*f1 *#,

"+ ).%$5(#)* /* 1%."$*#; ?.(*+,x *,y /*"0 &!&)*+,# 6"*!'.+6"*# /*G; A+ 4 4!.%# 7f(f−1(x)T f−1(y)) = f(f1(x))⋆f(f1(y)) =x⋆y; B2.Cf1(x)T f1(y) =f1(x⋆y)*,f1*#, /.+' "+ (#.).%$5(#)* /* 1%."$*#

/* G #"%G;

"#$%&'&(% )* *+) ;')#5%" G=G7 %! &'('(')*&+#(" "#$ 8**"6< '()"#"$%& !#' "$ #=+6 "#$ ,+-".$+/7 '!

*8)6")8 2=*+,"#"$%& !#'0 1234.53 )* *6

D E24$$!('4,(.+f : (Q,×)→(Q+,×) /&F+(* $4%f(x) =kxk *#, "+ ).%$5(#)* /* 1%."$*# 7 f(x×y) =kx×yk=kxk × kyk=f(x)×f(y)-∀x, y∈Q

D E24$$!('4,(.+f : (R+⋆,×)→(R,+)/&F+(* $4% f(x) = lnx*#, "+ (#.).%$5(#)* /* 1%."$*# 7 f(x×y) = ln(x×y) = lnx+ lny=f(x) +f(y)-∀x, y∈R+⋆;

D ?.(, a ∈ R; E24$$!('4,(.+ fa : (R,+) → (R+⋆,×) /&F+(* $4% fa(x) = ax *#, "+ (#.).%$5(#)* /*

1%."$*# 7 .+ 4

fa(x+y) =ax+y =ax×ay =fa(x)×fa(y)-∀x, y∈R;

D E24$$!('4,(.+f : (R,×)→ (R,×) /&F+(* $4% f(x) = x2 $."% ,.", x ∈R *#, "+ 5.).).%$5(#)* G

*+ *H*,

f(x×y) = (x×y)2=x2×y2 =f(x)×f(y)-∀x, y∈R;

D E24$$!('4,(.+f : (C,×) → (R,×) /&F+(* $4% f(z) =|z|$."% ,.", x ∈C *#, "+ 5.).).%$5(#)*- '4%

f(z×w) =|z×w|=|z| × |w|=f(z)×f(w)-∀z, w∈C;

D E24$$!('4,(.+f : (R,+)→ (R+⋆,×) /&F+(* $4% f(x) = ex $."% ,.", x ∈R *#, "+ (#.).%$5(#)*- '4%

*!!* *#, =(>*',(9* *,

f(x+y) =ex+y =ex×ey =f(x)×f(y)-∀x, y∈R;

D E24$$!('4,(.+f : (R,+)→ (C,×) /&F+(* $4% f(x) = e2iπx $."% ,.", x ∈R*#, "+ 5.).).%$5(#)*- '4%

f(x+y) =e2iπ(x+y)=e2iπx×e2iπy =f(x)×f(y)-∀x, y∈R;

,-(.(/&'&(% )* *++ ;')#5%" G *'##>2" %! <6<("!$ !"%$)" e7 6=+(84" &'('(')*&" f(e) = n "#$ !"%$)"

28!# f(G)0 1+ 68 6'+ T 2" G "#$ 8##'.+8$+:"7 86')# ."66" 2" f(G) 6="#$ 8%##+0 1+ G "#$ %! 4)'%*"7 #'! +(84"

&'('(')*&"f(G)"! "#$ 8%##+ %! "$f(x1) = [f(x)]1 ? 6=+!:")#" 2"f(x)28!#f(G) "#$ 6=+(84" 2" 6=+!:")#"

2" x 28!# G0

Références

Documents relatifs

Actuellement, outils et ateliers sont utilisés très largement, mais avec un sens plus large (et parfois abusif) donné au terme de génie logiciel.. L’industrie représente encore plus

Le congr` es a ´ et´ e tr` es bien pr´ epar´ e tant au niveau des conf´ erences, qu’au niveau culturel.Grˆ ace aux diff´ erents expos´ es nous avons pu mesurer l’importance

La structure d'ensemble ordonné (ensemble muni d'une relation d'ordre) a été mise en place dans le cadre de la théorie des nombres par Cantor et Richard DEDEKIND (1831-1916)..

La structure d'ensemble ordonné (ensemble muni d'une relation d'ordre) a été mise en place dans le cadre de la théorie des nombres par Cantor et Richard DEDEKIND (1831-1916)2.

Une matrice A qui a une valeur propre d'ordre de multiplicité m &gt; 1 mais qui n'a pas m vecteurs propres linéairement indépendants associés à cette valeur propre est appelée

Les quantités de chaque facteur primaire intervenant dans la fabrication de chaque produit ni peuvent être rassemblées dans une matrice P , dont les lignes se rapportent aux

[r]

Montrer que SL 2 ( ℤ ), muni de la loi de multiplication de matrices, est un groupe non-ab´