1 2 n π =, 1π 2π nπ. ϕ = = {(1,3), (2,1), (3,3)}

!"#$$%&' ()*+,-./ 0/& 0/),0 )))1 234546' !"#$% %&'#( 789:8 ;< =>8?8' )*'+!% $,",-. (/,"0! 123145 &@A5=9' --!"*$%6-!$7589(/,:,0578 7BC95D' ;;;5-!$7589(/,:,0578<∼--!"*$%< !"#$%&'(')

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

=8$*-!$%&8(!B ,'?:$,',B H,:+,$ &!F !/. !&% F8(&A/%&. F8(&A$ 7*/ ! $!B$,",-L,. @!:+ -A($#' LA"-%'+,B -#0*B (,:M$(?:,$ &!F @!:+ -A(B!& (,:M$(?:,$ B+VH$5 =/ ,',&$"*B%&8( $,(/$,&,$ KBA''#!B. (,:M$(?: B?'&W' &,'' -,:*'0!B%. 0, %$$ M"*$$ @!:+ ,',&$"*B%&8( H,:+/,$ 7!(/BA'7!$#5

!"#$%&'()* +,-!. /01#0234 !"#$%&'(&)"$*+ ,--./ 0!123&2 4,/5 6$27*8 689"7'

:;;!9 $) !"#$%&';$9 $ 2*>'>?!) $ 9@(!*9!)# 0A=$"<$9$* 9!"" *3%;8B 8'<>*">' ;>"9CD "8 ($28&E87+ "!9>F!)>'+ 8;%3"&'8 >' >29!)>'8 ?$"<$)+ ')C2G!9*H(+ 8;G!9*H(+ 18G!9*H( >' 8%!;*893' "!9>F!)>'+ "!9>F!)>'!9 ')A2)$*$ >' 8;(!2)!+ E'AFA2*+ I1!"DE'AFA2*+ $'')AE8$*H(+ 8""!*(! 9A<<3D *$*H( <J(!"!*+ !=K'>=!"!<+ E'AFA2*!"!< 8;(!2)! >' ?$*(&;K$+ %!*!2<8;&;' >' *3"$G%A;'&=$8/ I) !"#$%&'?A) $G&;"A** G!=K)!*B

• •

!"##$%&'# (# !%)*+,%# "!-*.,"+ NA"K=A; L8$%7+ O)!=!%+ 4PPP/ O)!;%2!8 Q=;!'B /%#$&,(0 1"0*1"0%&"+ NA"K=A; L8$%7+ O)!=!%+ 4PPRS,--,/ L"39A(8*' M$GA'B

! "#$%&'()! T!*')#"!=!'

I)

A

?$"<$) F!2<3*&E878; $

n

FA)8*H( !=>')2! $)

I

π ∈ Sn

{1, . . . , n}

π : A → A

18G!9*H( "!9>F!)>'!9!* >2*GC;9/

?$"<$) @'')!' F!2<3*&E878;$9 ?$"<$)&*

Sn D;!"

G!"@"GC9/

*! +#,-,./!

F!2<3*&E87* *'A2A' H2&'<7%%$"

π=



1 2 ··· 1π 2π · · ·

 n , nπ

($=K !"!;*B

π = {(1, 1π), (2, 2π), . . . , (n, nπ)}.

0! 1.,23!

U$

α ∈ S3

$) $ F!2<3*&E87+ $
α=

4! 1.,23!



1 2 3 2 1 3



1α = 2+ 2α = 1

>'

3α = 3+

$99A2

= {(1, 2), (2, 1), (3, 3)}.

V!< <8;%!; "!9>F!)>' F!2<3*&E87+ F>"%&3" $

ϕ=



1 2 3 3 1 3

"!9>F!)>' '! ;!< 8;G!9*H( 29!)>'8 ?$"<$)

2



= {(1, 3), (2, 1), (3, 3)}

1 >' 3 !"!F!5 '! ;!< ')C2G!9*H(


!"<>;!9 ;8;E'!; #'!5/

5! 6.7#,! |Sn | = n! 8! 1.,23!   1 , S1 = 1     1 2 1 2 , , S2 = 2 1 1 2             1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 S3 = , , , , , . 1 2 3 2 1 3 3 2 1 1 3 2 2 3 1 3 1 2

9! 6.7#,! (Sn ; ◦) :! 1.,23!

2#343,05

O)&
α=



 1 2 3 2 1 3

>'

β=



1 2 3 2 3 1



!"#$%&'()* +,-",.%&%/ 0$12$*3 4-56 #(71!7 x !8!#"! x(αβ) = (xα)β 9!, . 8!*: !,:+ +,-",&+ 1!;7<'()2.=/ 0!4&% 1(αβ) = (1α)β = 2β = 3, 2(αβ) = (2α)β = 1β = 2, 3(αβ) = (3α)β = 3β = 1,

.,., αβ =



 1 2 3 . 3 2 1

>-+% *(+,&#-82$* . βα +,-",.%-% (+ 9. ,&")2!8!* !84.56&+&?.8=@ 1βα = 2α = 1, 2βα = 3α = 3, 3βα = 1α = 2,

.,.,

  1 2 3 βα = . 1 3 2

A!56B* :+,"!3 4-56 αβ 6= βα3 .,., . !"#$%&'()* +,-",&+. 7!# *-##$%.%(?!8 β=

!,:"% β

−1



1 2 3 2 3 1



= {(1, 2), (2, 3), (3, 1)}

= {(2, 1), (3, 2), (1, 3)} = {(1, 3), (2, 1), (3, 2)} =

0!"#:+,!%!+!7 β :+ β −1 +,-",.%. ., (1!7%(*$+ 8!*: !,:+@ ββ

−1

=β

−1

β=





 1 2 3 . 3 1 2

 1 2 3 . 1 2 3

! "#$%&'()! C π ∈ Sn !"#$%&'() ., x ∈ {1, . . . , n} !8!#!% #-,5.%2.3 4. xπ 6= x/ C π ∈ Sn &8%.8 #-,5.%-%% !8!#!* 4.8#.,&% Mπ D?!8 2!8E82B*3 .,., Mπ = { x ∈ {1, . . . , n} : xπ 6= x }.

*+! ,-./0! C,

  1 2 3 α= 2 1 3 !"#$%&'() &8%.8 #-,5.%-%% !8!#!* 4.8#.,. Mα = {1, 2}/

**! 1-2/-3#4! F&76 -86.7 π ∈ S9 !"#$%&'() ?.73 .#!86"! (1) Mπ = {2, 3, 5}3 (2) |Mπ | = 13 (3) |Mπ | = 23 (4) |Mπ | = 3G *5! "#$%&'()! C π, σ ∈ Sn !"#$%&'()*.% (1!5!77!* 7!?!,,B*3 4. Mπ ∩ Mσ = ∅/ *6! 1-2/-3#4! (1) C, Sn 4.8#.,-7 ., H(1!5!7+:5I "!8&'() "!J!K
! ! π, σ ∈ Sn "#$%&'()*+, *-#.#/#,0 (1) πσ = σπ 0 23 (2) (πσ)k = π k σ k %*/-#/ k #.234$#5

*7! 8-9#.!

!,,1$

!" #$%&'()*" !"#!$ n ≥ k ≥ 2% &' () a1 , . . . , ak ∈ {1, . . . , n} !*!+!, -./0$,&$1 ,2*3$435

)6!,7 8,,0/ ()1 ( π ∈ Sn -!/+91.:;<1% (+!*#/!

a1 π = a2 , a2 π = a3 ,

77 7 ak−1 π = ak , ak π = a1 ,

&' xπ = x +;$=!$ x ∈ {1, . . . , n} \ {a1 , . . . , ak } !*!+/!% :;,*9'$(, $!>!))2, &' /3>;=!$ ?"# @!*3*@2,A B k ').+01 ( :;,*9' C0'').$(, C?>@9,7 +" ,-./0" B)

π = (a1 a2 · · · ak ). $!>!))2,7 B 2 C0'')D'."D

:;,*9'0,(1 1/($')-0)?:;<,$(,

 1 2 3 2 1 3 -!/+91.:;< :;,*9'% +;>!* ( k = 2% a1 = 1 &' a2 = 2 >.*(')1.''(* &--!$ !)1 ,(-@9,% ()() α = (1 2)7 E;>!* α C0'')( &--!$ 2% !)&/1 α 1/($')-0)?:;< ;'7 B   1 2 3 β= 2 3 1 α=



( -!/+91.:;<1

-!/+91.:;< ');$1&$ :;,*9'% &' β = (1 2 3)7 1" 2-3/-4$5"

E; () (a1 a2 · · · ak ) :;,*9' .*1(* +0)"(1011 !*!+!, C(*+()(F G"()5!% C0"# C( π, σ, τ ∈ S7 -./0$,&$1 ;=!"!$ -!/+91.:;<,% (,,0/ (πστ )5 = π5 σ5 τ 5 F 6" 7$89$8:;-4" H!"#2, &')/!% C0"# !"# -!/+91.:;< :;,*9'0' (*(,4($ >(*< +!"(=.'( $!+ !"#&/1!*+IJ 8"#/&')1 9"#($()1 ( -!/+91.:;<1 1344K&*!,&--!$ ;' K!*?/C(1@9, :;,*9',&$1A (1) (2)

(1 2 3) = (2 3 1) = (3 1 2).

B +.';, -/04*&+( -!=;" ()% C0"# () (1 2 3) -!/+91.:;("# !'!1*!" () S4 :'0-0/1 !*!+!5!7 L!/+&')!1!'!$ C( S3 54!*; -!/+91.:;<,/<* 4!')&*2$,% (,,0/  

S3

1 2 3 , 2 3 1

(1 2 3) =

>;')0$1 S4 54!$ +./

(1 2 3) =



 1 2 3 4 , 2 3 1 4

&' !) ( ,&1 -!/+91.:;< $!+ 9"#($()7 M"#($ !) ( -/04*&+( () ;=!$1;,9' -!/+91.:;< N idO @!*3*&'&>!* ;'% (//<* '!+ *!C!1 !*=3$1!$;% C0"# +!*#;, -!/+91.:;<:'0-0/14($ C(')$.*@9,7 <" ,-./0" S1 = {id}, S2 = {id, (1 2)}, S3 = {id, (1 2), (1 3), (2 3), (1 2 3), (3 2 1)}.

=>" 2-3/-4$5" (1) (2) (3) (4)

P.$# 1/($')-0)?:;< >($ S4 54!$F P.$# Q5C0'')D'."D :;,*9' >($ S4 54!$F P.$# R5C0'')D'."D :;,*9' >($ S4 54!$F P.$# S5C0'')D'."D :;,*9' >($ S4 54!$F

!"# $%&'() *+" S4 ,-."/ !"# 0'#+" 1.23(4!$%5 *+" S4 ,-." +3.'# ".3 $%&'()/ !"# n,60))78)!98 $%&'() *+" Sn ,-."/ #$%&'" :.23;)7.4.)." ".3 3%"<." 1.23(4!$%5 $%&'()= *.9#>& 1;'
(5) (6) (7)

!"

π=



 1 2 3 4 5 2 3 1 5 4

1.23(4!$%54? :.9#>& @.'= 609# π $%&'()= ;) 4.&%"4)>& +74 +7 .).4.4= +3%&02 a1 = 1? A&&02 a1 π = 2= +7+7 a2 = 2= 40*!--! a2 π = 3= +7+7 a3 = 3? B &C*.4&.7D ';1;)-." +74 &+1E(&= 609# a3 π = 1 +3% ;11." .9#."'D a1 ,9#.'= +7+7 k = 3 ;) +7 (1 2 3) $%&'()4 &+14(&? F%)70"4 π 4C-- .'.3.4 3079+4 3%"4 3= 4.6!4 π ".3 .9#."'D (1 2 3),3+'= +7+7 a1 6= 1? G%"<." 3!)

.).4-." 6+)0"'5 .''."430"
π = (1 2 3)(4 5).

" ($)*%" !"#$" Sn %&$'! ($)*+,-.!/ $'0-'' (-)1"23", !#$4$" .!2'+512 561)67,723",8 35 $6 76 $'0-''9,-5 7 ,3":$602 51))$"#;3,0' $',$2!",<$ $4:3),$'*=$" *$4>7,-)161,,? @A6 !#$",!2+5 ($)*+,-.!/, .!2'+512 B)$5 561)67,-"72 ,$2!",;B2?C

+" #$%&'" B<E(& 3.9 + π = (5 2 3 4)(1 3 5)(4 3 7) 1.23(4!$%54 1!20"&;"4 %<.9." $%&'()0& )7027+4+&;"4? :.&%"4)>& +70&+4 +7 .'.3.&.4= 3.'#.&.4 + )7027+4 *+'+3.'# 4+9E+ 3079+4I {1, 2, 3, 4, 5, 7}? F.9#>"& &% .7.& &C7>' .9#.4= 30"<E(& +7 1,9#.4= ;) )7!30'E(& &%= 609# .74 + π 1.23(4!$%5 3%'#." .'.3.&-. *%)7% !4I 1π = 1(5 2 3 4)(1 3 5)(4 3 7) = 1(1 3 5)(4 3 7) = 3(4 3 7) = 7.

J0'#4+))(& + &+1044 .'.3.&&.'= +7+7 7π = 7(5 2 3 4)(1 3 5)(4 3 7) = 7(1 3 5)(4 3 7) = 7(4 3 7) = 4, 4π = 4(5 2 3 4)(1 3 5)(4 3 7) = 5(1 3 5)(4 3 7) = 1(4 3 7) = 1.

F%))7+;24>"& +6607 +7 .'.36.7= +3%-D' &%%"<('4("&= 4.6!4 3.9*+" +7 .')D $%&'()("&I (1 7 4)? B 3+2+<;& .'.3.&-D' *.9#>& + &C*.4&.7D4= 30"<E(& + 2,4D4= ;) )7!30'E(& &% 609# .74 π 3%'#." .'.3.&-. *%)7% !4I 2π = 2(5 2 3 4)(1 3 5)(4 3 7) = 3(1 3 5)(4 3 7) = 5(4 3 7) = 5, 5π = 5(5 2 3 4)(1 3 5)(4 3 7) = 2(1 3 5)(4 3 7) = 2(4 3 7) = 2,

+7+7 + 3!)0<%& $%&'() + (2 5) 42+")7107K$%5? L%3+2+<4 3;9 + 3= +3.'#2. .'*;9.7*. + )7!30, '!)4 +74 &+1E(&= 609# 3π = 3(5 2 3 4)(1 3 5)(4 3 7) = 4(1 3 5)(4 3 7) = 4(4 3 7) = 3,

+7+7 π + 3,3+4 ".3 3079+4E+= 4.6!4 .74 +7 .'.3.4 M9#.'3." &K*>' 6+9#6+4E(&? :.6!4 π 1!20"&;"4 %<.9." $%&'()0& )7027+4!2+ -0"4044 +'+&E+ π = (1 7 4)(2 5)? A74 + )7!30'!)4 ".3 K2E(& '. !'4+'!-+"= 6+".3 @.E-." *;9.77>& .'N ," -$.&$/*0" !"# 0'#+" 1.23(4!$%5 *+" G,-."= +3.'#".& 1!20"&;"4 %<.9." $%&'()0& )7027+4!2+ -0"4044 +'+&E+ P +'+&8I (1) G = S4 = P = (· ·)(· ·)= (2) G = S5 = P = (· ·)(· ·)= (3) G = S5 = P = (· ·)(· · ·)/ 1" ($)*%"

D$,560'$4$5 8

π = (a1 a2 · · · ak ) ∈ Sn (1) π −1 = (ak ak−1 · · · a1 ) (2) π k = id (3) i ≡ j (mod k) πi = πj

E7

8 8 7221)

?

.!2'+5)7

!"#$%&'()* +# ((1 2 3 4)(5 6 7)(8 9))−22 ,-.%)/$0!1/ ,$.&2*32/ !4-5-2 0!*')"&* "#&.#+/+*32/6 7!8-' +# (1 2 3 4)9 (5 6 7) 3" (8 9) 0!*')"&* ,$.&2*32/ !4-5-2-*9 -#3./ !" #$%&'"

((1 2 3 4)(5 6 7)(8 9))−22 = (1 2 3 4)−22 (5 6 7)−22 (8 9)−22 .

:# (1 2 3 4) 0!*')" ;&""#+ 4 3" + −22<-4!* ;+/8$2=$/ *-.-"">*6 7!8-' −22 ≡ 2 (mod 4)9 -#3./ ?+"&2'1+2

(1 2 3 4)−22 = (1 2 3 4)2 = (1 3)(2 4). −22 ≡ −1 (mod 3)9 !''-/8- −22 ≡ 0 (mod 2)9 +#+# (5 6 7)−22 = (5 6 7)−1 = (7 6 5), (8 9)−22 = (8 9)0 = id.

@-;$/ (" #$%&'"

3"

((1 2 3 4)(5 6 7))−22 = (1 3)(2 4)(7 6 5).

A'4()* %-5 +#

(1 3 2)(2 5)π(4 5 7) = (2 6)

-5=-2'-/-/6 :# -5=-2'-/ %!24*3/ &'4+'$/ )5=+2+##+' + ,-.%)/$0!18+' )5=+2+..1' +# &'4+'.1' B-"#&.&#;+/()*6 C'D"#E. B+'.1' "#&.#)2* (1 3 2) !28-.#38-'F (1 3 2)−1 (1 3 2)(2 5)π(4 5 7) = (1 3 2)−1 (2 6),

+#+#

C#/ G&'=/+/8+ +#/ *+,()*9 ;&5=

(2 5)π(4 5 7) = (2 3 1)(2 6). π = (5 2)(2 3 1)(2 6)(7 5 4),

+%!/ + "#&*$"&" %14&2 ,$.&2*32/ !4-5-2 0!*')"&* "#&.#+/$.+ B&2/)2*F π = (5 3 1 6 2 4 7)6 )" *$+,%"

!"#$%&!'!# ()*&+# ,!&-./0"1 ".02#$34$-()1* #$4.$0"0*52"6 75'3!8)' (a1 a2 a3 · · · ak ) = (a1 a2 )(a1 a3 ) · · · (a1 ak ).

9:;!"*!$5#*533!26 7)28!2 3!.7+"<()1 ".02#$34$-()1* #$4.$0"<.0 =42"/0"1 >8! !$ <&"0&<=02 2!7 !'?5."!&7@AB

-" #$%&'" (1 2 3 4)(5 6) = (1 2)(1 3)(1 4)(5 6)9 4- %!8-' (1 2 3 4) = (2 3 4 1)9 -#3./ (1 2 3 4)(5 6) = (2 3)(2 4)(2 1)(5 6)9 8+5= (1 2 3 4)(5 6) = (2 3)(5 6)(2 4)(2 1)9 %-./ !4-5-2

/.+2"#,&#H0!1* G-'0"-.3';-/D*6

./" *$+,%" C)28!2 3!.7+"<()1 ;0'? (#0* 3<.4# ;0'? (#0* 3<.0"&02 #4* ".02#$34$-()1 #$4.$0D "0*52" -./0"1 ,!&B

: π ∈ Sn ,-.%)/$0!1/ ,$.&"2+* 2-8-##>*9 ;+ G-'B&2/;+/1 ,$.&" "&* /.+2"#< ,&#H0!1 "#&.#+/$.+6 : 2-%,$.&" ,-.%)/$0!1*+/ ,$.+/'+22+* 2-8-##>*6 @&8$BB$ 4-I2!$'()*F ( +1, ;+ π ,$.&"9 sgn π = −1, ;+ π ,$.+/'+26 . " 8$9&$:,;" :# +'$BB! $''H/$"&* *E#>' %-'=-* !5+#+* 3" %-'=-* ;+%!"+*J (1) :# !4-2/!/$" ,$.&"6 (2) 7!24-2 /.+2"#,&#H0!1 ,$.+/'+26 (3) 7!24-2 ,$.&" ;&""#K 0!*')" ,$.&"6 (4) 7!24-2 ,$.+/'+2 ;&""#K 0!*')" ,$.&"6 (5) L$.&" ,-.%)/$0!1* "#&.#+/+ ,$.&"6 (6) L$.+/'+2 ,-.%)/$0!1* "#&.#+/+ ,$.&"6 (7) L$.&" 3" ,$.+/'+2 ,-.%)/$0!1 "#&.#+/+ ,$.+/'+26 (8) L$.+/'+2 ,-.%)/$0!1* !28-.#- ,$.+/'+26

.0" 1,234567"

!"#$%&%'$() *+", & 4 × 4-",!. %+/+"&%0. '1%2(3&4 & 3&/+/5&/6 (!7581//1.30/ 4!# /!*!% !/81//9%&46 & '+33+/5&/6%: ; < = > < ; = > ? @ A B ? @ A B B= A= C ;D ;; ;< C ;D ;; ;< ;= ;> ;? ;= ;> ;? E '1%2( #645!4 1//1.1*+7 *+771F!45!/'G( &7 S16 H.+I+F% !",6( !/!#2%) #2"I!56" J",) *+", &7 GF!. #!78 *!/,23! & 16-+. .71#+% (2I7!/'G() 2. & (&I+%% ! "#$%&!

a1 a2 a3 a4 a5 a6 a7 a8 T = a9 a10 a11 a12 a13 a14 a15 a16

%13/17&%+%K!/*&.741/L& (2I!77G( & πT =

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16



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πA = id  I1F+.) #9" & '+33+/5&/6 1//&I+%F& πB = (1 2)  I1F&%/&4M O!*1% 4!# /!*!% &7 A 1//&I+%30/ & B 1//&I+%3& '$%46M '! ()*+,-./! E7 Sn H.+I+F%+% &7 n-!5F!45T .76##!%F6($. H.+I+F%4&( 4!L!77G(M E I1F+. I!F#$%1H60( An = { π ∈ Sn : π I1F+. } *&/#&7& .764%24 H.+I+F%+% &/(+%) &#!/,4!( 4!L! &7 n-!5F!45T &/%!F41/0 H.+I+F%M 0! 1#2%#3)4! (1) (2) (3) (4)

U14, I1F&%/&4 I!F#$%1H60 L&4 S3-3&4V U14, I1F+. I!F#$%1H60 L&4 S3-3&4V U14, I1F&%/&4 I!F#$%1H60 L&4 S1-3!4V U14, I1F+. I!F#$%1H60 L&4 S1-3!4V

5! 6#7)$!

!"#$%&!'!# n ≥ 2 !'(#$)! |An | = n!2 *

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

#1#1 567898:1

! "#$%&'()*+, .'/0 123 04 03&55)

+;: +4;'4080:=<8@ 

1 7  1 (2) β = 1  1 (3) γ = 4

2 4 2 6 2 5

(1) α =

#1;1 567898:1

3 2 3 5 3 7

4 3 4 2 4 1

5 6 5 4 5 2

S7 6523)

6 5 6 3 6 6

72'%*8&()9:08 7&';<:=<8 )>2?2< ():3*6

 7 A 1  7 A 7  7 $ 3

B>/0 %2? 0 :CD28:24E

S7 6523)A

7&';<:=<8 )>2?2< ():3*+;: +4;'6

4080:=<8 23E&33F8;88 72'%*8&()9:08 :=8+;';+ F'&+%9>50<@

(1) δ = (1 3 6)(2 7 5 4)A (2) ε = (1 7)(2 6)(3 4 5)A (3) η = (1 5 4 2 7 3)$

#1<1 567898:1

B4 23E4E :=8 1230>0850< 52D2428288 72'%*8&()9:593 :))<>*3D0

0>/0 %2? 04 03&55)

S7 6523)

72'%*8&()9:08 7&';<:=<8 )>2?2< ():3*+;: +4;'40806

:=<8@

(βα)−1 ,

#1$1 567898:1

αn (n ∈ N),

εη −1 βγδ −1 ,

B>/0 %2? 0 :CD28:24E

S9 6523)

(ηβ)2006 δ 2007


−1

.

72'%*8&()9:08 7&';<:=<8 )>2?2<

():3*+;: +4;'4080:=<8@

(1) α−1 βα


109

A 0G;3



 1 2 3 4 5 6 7 8 9 , 3 5 4 6 7 1 9 2 8   1 2 3 4 5 6 7 8 9 ; β= 7 8 9 2 5 1 6 4 3 −1  (2) (1 5 4 3 7 2)9 ((2 9 3)(4 5 2 7))120 (4 8 1) $ α=

#1=1 567898:1

H2'2++2 %2? %)<>04;:08 0

σ ∈ S8

72'%*8&()9:08A 0%23I2:'2

823/2+J3A G;?I

((1 2 3 4)(7 3 8))3 σ(3 4) =

#1>1 567898:1



−1 1 2 3 4 5 6 7 8 . 8 5 6 4 3 1 2 7

KC<8+2 23A G;?I 04 23+E :=8 1230>0850< 52D2428288

α, . . . , η ∈ S7

72'%*8&()9:A D030%)<8 0 +2?F8+=?J::23 %2?0>;88 03&55) 72'%*8&()9: 7&';+0:62 D0?I 7&'0830<;:@

η −1 δ 112


111

,

(εγα)−1 β −1 δη 9


2

.

!"#$$%&' ()*+,-./ !0/1!0/),0 )))2 &30,4-50/ 6 7782 !9-:);* <2=

!"! #$%&'&(!

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

(1) γ k = id' (2) γ ℓ = 6 id' (" ℓ ∈ N

!)! #$%&'&(!

3+/.66

k

($//#0/1!0

γ ∈ Sn

k 7,1%8 π ∈ Sn 9.5*4:12+;5" π n! = id8 k ∈ N /#1*$:' "*.%)5. π k = id=

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

%.(.: *.!"-,+ " %.!3+/.66 $%)",

2+3%4/5"

<$!)",

!*! #$%&'&(! (1) (2)

"# "#

>4:"//" *.!' ($!) S9 *+,-., .%.*. .%?1%% (1 2), (1 3), . . . , (1 9) :5",/#9$#@2+;3 /#$5#":"3A,:B (1 2) A/ (1 2 3 . . . 9) 2+3%4/$3 /#$5#":"3A,:8

! +! #$%&'&(! :.%&./D%' ($!) 2

C.5.//. *.! *+,-"#$3": "

(1) π = (1 2)B (2) π 2 = (1 2)(3 4)8

π ∈ S6

9.5*4:12+;3":' "*.%).35.

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

!"#$%&'(&)"$*+ ,--./ 0!123&2 4-/5 6$27*8 689"7'

:;;!9 $) !"#$%&';$9 $ 2*>'>?!) $ 9@(!*9!)# 0A=$"<$9$* 9!"" *3%;8B *!'*+ *!'* $%%8*C( >' <3"*8D"89$*C( E'ADA2*F$+ ($"7' ')&' *3"$F%A;'&=$89/ G) !"#$%&'?A) $F&;"A** F!=H)!*B

• •

I"39A(8*' J$FA'B L)!;%2!8 M=;!'B

! "#$%&'()!

!"##$%&'# (# !%)*+,%# "!-*.,"+ KA"H=A; I8$%7+ L)!=!%+ 4.../ /%#$&,(0 1"0*1"0%&"+ KA"H=A; I8$%7+ L)!=!%+ 4..NO,--,/

G ($"7' ')&
')&
C

*! "#$%&'()!

(a, b)

G)

>'

(c, d)

C = R × R/ 9A' ')A2)$*$B

(a, b) + (c, d) = (a + c, b + d)

+! ,-./0!

(a, b) · (c, d) = (ac − bd, ad + bc). G)

(1, 2)

>'

(3, 4)

9A' ')A2)$*$B

(1, 2) + (3, 4) = (1 + 3, 2 + 4) = (4, 6),

>'

(1, 2) · (3, 4) = (1 · 3 − 2 · 4, 1 · 4 + 2 · 3) = (3 − 8, 4 + 6) = (−5, 10).

1! 2-3#.! (C; +, ·)

0*#02

3%$4)560+#7+$!"02 68;%!; 9@;;H!; "!!""!;#28)?!*#+ ?$ $) $%%8*C( !=H'>=;!9 $ (0, 0)+ =;!9 $)

(1, 0)

9A2%!9!' 9>2%>'

$ <3"*8D"89$*C( 8;(!2) ">*!)>'!B *!*')#"!=!'+ $) $%%8*C( !=H'>=*#" 9Q"@;1@)#

(a, b)−1 = <8(!"

(a, b) ·

4! ,-./0!



−b a , a2 + b2 a2 + b2



=



(1, 2) = (1, 2) · (3, 4)−1 = (1, 2) · (3, 4)

5! 6-7/-8#9!



a −b , 2 2 + b a + b2

3 −4 , 25 25



=



3 − (−8) −4 + 6 , 25 25

G 9@(!*9!)# &""C*&'A9 9@)Q"
8%)9*) a, b ∈ R *#*0()

(a, 0) + (b, 0) = (a + b, 0),

;! "#$%&'()!

a2



a2 −b2 −ab ab − 2 , + 2 2 a +b a + b2 a2 + b2 a2 + b2

(1) (a, b) · (c, d) = (a · c, b · d)+ (2) (a, 0) · (c, 0) = (a · c, 0)+ (3) (0, b) · (0, d) = (0, b · d)R

:! 2-3#.!



(a, 0) · (b, 0) = (a · b, 0), S!*')#"!=!'



(a, b) ∈ C 8;(!2)!



=

= (1, 0).



11 2 , 25 25

a, b, c, d ∈ R





.

!'!*>;B

−(a, 0) = (−a, 0),

(a, 0)−1 = (a−1 , 0).

a ∈ R !'!*>; $) (a, 0) 9A
>' ;!< 8' 9Q"@;1@)*!*FQ9
(0, 1)

9A
#$+1 *-5(,0*!1? 1@94) *!;+!! a + bi "!"&.")2 :47+..+ i2 = −12

z ∈ C !"#$%&' ()*# a + bi +%+!,+- .+%/ 0&%12*(*3 z !+-"-4!5( +%+!6*-+! -&.&))7!8 ) a ∈ R ()*#"3 z .+%/( 29()9-&!: #1; + b ∈ R ()*#"3 z !9$)&3&( 29()9-&! <1.65!: 9( a = Re z: 4%%&3.& b = Im z=.&% 6&%>%67!8 ) i !"#$%&' ()*# -&.& !9$)&3&( &;?(9;8

!" #$%&'()*"

" +,-./" !>.&3!&)@ ()*#"%*(,+- A(+! +)3 <+()-*%35! !4: <";? C 3&(3 B+)+) 92.9-?&(&! + ()"!*("( ()*#"%*(4 ()+,*%?"!C 9( i2 = −1D (a + bi) · (c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i

E&((7! >(()& + !+$"33 &2&F#9-?3 + !"#$%&' ()*#"! ()"2)*(*-+! F&G-1A4/6*.+%H #5%34$%4= !+31. 4-.&2) !4()*#"%*(*-*% +)3 + 6/% 4(#&23 +)"-"((*;"3 +%!+%#+))5!: <";? (a + b)(a − b) = a2 − b2 D (a + bi)−1 =

1 1 a − bi a − bi a − bi a −b = 2 = 2 + i. = · = 2 a + bi a + bi a − bi a − (bi)2 a + b2 a + b2 a2 + b2

I&;?&- +F"33 + (1!,+- &;? J&(A+23&(=09%& F&29!()>;K !""2F4-*3+2&-F()&2: 9( 0&%&%3&((7! #&; +) a + bi !"#$%&' ()*#-+! +) (a, b) !""2F4-*3*6L $"-3"38 M;? !+$65! + !"#$%&' ()*#(1!"3: #*( -9.&- + N+5((=09%& ()*#(1!"38 ) &%(@ 3&-;&%?3 B+,()A4(()+C .+%/( 3&-;&%?-&!: + #*("F4! 3&-;&%?3 B"2F4-*3+C $&F4; !9$)&3&( 3&-;&%?-&! <1.65!8 .+%/( 3&-;&%?&- 3+%*%<+3/! + .+%/( ()*#"!: + !9$)&3&( 3&-;&%?&- $&F4; + 34()3+ !9$)&3&( ()*#"!8 1" #$%&'()*" z = a √ + bi !"#$%&' ()*# !"+,-.*%/,*+ 0 z = a − bi !"#$%&' ()*#"/1 2( 03()"%4/ 25/2!2+ 0 |z| = a2 + b2 60%7( ()*#"/ 25/,8!9 !" #$%&$%'()*" !"#$%&' ()*#(:!"+ 0 !"+,-.*%*( +&# #*(1 #;+/ 0 60%7( /&+.&%<5& 60%7 /8!5=)2(1 0) 03()"%4/ 25/2! 0) "5;.7/7% >+-%%*/7%? #25/ /*6"%(*.1 0 !"#$%&' ()*#"! =(()&0@*(0 $&@;. >A&%
(4) (5) (6) (7) (8)

u·v =u·v u/v = u/v , -+ v 6= 0, u = u ⇐⇒ u ∈ R, u + u = 2 Re u, u · u = |u|2 .

!"#$%&!'!# u, v ∈ C #$()*+ (1) |u| = 0 ⇐⇒ u = 0, (2) |u · v| = |u| · |v|, (3) |u/v| = |u|/|v|, -+ v 6= 0, (4) |u + v| ≤ |u| + |v|, (5) |u| = |u|. 0" ,)-$." /!'0!1!2 z1 , z2 , . . . , zn 23)4&!5 #$()32 6'0, -3'0 + 23)4&!5 #$()#7231 +$ (&"+8 &92 )!'-+"(*3$3"" 43&:'31 231;!5, <# + z1 , . . . , zn =#6=#32 +$ >*+)9"+"> ?(*(#(;+& !&&!1"<"!# :*(10@+1 -!&0!$2!A1!2 !&. B223* + 43&:'31 "!*C&!"! + 2D;!"2!$% 2<4&!""!& #$()3&-+">E /" ,)-$."

1 Im(z1 z2 + z2 z3 + · · · + zn−1 zn + zn z1 ). 2 1" 2$345678" B.< +&#+-%%0 z !"#$%&' ()*# 05.-#&+/-#0 0) 0 ()=.1 0#;6&% 0 60%7( /&+.&%< $");/:6 C&%2/ &% !&%% C"5.0/+; 0) "5;.7 !=58%1 A".< */#&+,&+ 0 zD+&! #&.C&%&%E $"+/"+1 0#;/ arg z D6&% ,&%=%,8!9 +-%%0 ()*#+0! +;+F(&+ 05.-#&+/-#09

) 0%*33; *%%:/*("! !=)8% #&%<&! ;.0)0! 2( #&%<&! A0#;(0!G H;+@&+ +&#+-%%0 60%7( ()*# 05.-#&+/-#0 +-%%09 H;+@&+ π 05.-#&+/-#4 !"#$%&' ()*# 60%7(9

9" :);<)*$=" (1) (2)

! i "#$%&'( )!*$ +,-.$'/0.$+ 3π/21 ! 1 − i√"#$%&'( )!*$ +,-.$'/0.$+ −π/41 ! 12 − 23 i "#$%&'( )!*$ +,-.$'/0.$+ −π/31 23/4'/ /'$/.&&+ z ∈ C )!*$,+ arg z = arg z1 23/4'/ /'$/.&&+ z ∈ C )!*$,+ arg (−z) = arg z + π1 23/4'/ /'$/.&&+ z ∈ C )!*$,+ arg (2z) = 2 arg z1 #$%&'" !"#$%&!'!# 0 6= z ∈ C( r ∈ R+ )# ϕ ∈ R #$*+,- !#!"). z = r(cos ϕ + i sin ϕ) ⇐⇒ r = |z| )# ϕ ≡ arg z (mod 2π). )&*+,-./" /'$/.&&+ "#$%&'( )!*$#"

(3) (4) (5) (6) (7) (8)

!" ("

z = r(cos ϕ + i sin ϕ)

+&+"5+/ 6+&7 8'&9,*)*0 0,3-#/#$'0,3".) +&+"/+" /'6'!!:"1 /.&&+ "#$%&'( )!*$/+" /3/;)'/ 0,3-#/#$'0,3".) +&+"<+1 " 0&12&134$5" /.&&*07& ":&=/5=!> "#$%&'( )!*$#" +,-.$'/0.$+ ;)+" ?$#4.&# 2π @A +!+! 2π '-B)! )!*$C 0=55)!=,=)'30>& '&0'"3/06' $'-D+0*,#!#001 E!B,0 + "#$%&'( )!*$#" 0,3F −3π -#/#$'0,3".) +&+"<+ )'$ '-GB,0'&$HI %B&4*.& $3/4 cos π2 +i sin π2 A $3/4 + cos −3π 2 +i sin 2 +! i "#$%&'( )!*$ 0,3-#/#$'0,3".) +&+"<+1 J3)!#/0 D+ '-G "#/",B0 "#$%&'( )!*$ 0,3-#/#F $'0,3".) +&+"<*0 "'&& $'-D+0*,#!/./"A +""#, +! +,-.$'/0.$#0 $3/43- + [0, 2π[ 3/0',6+&&.$F 5+/ +4<." $'-1 6" #$%&'" !"#$%&!'!# ./&&*"0& -1&2.32$% u = r(cos ϕ + i sin ϕ) )# v = s(cos ψ + i sin ψ) -,+4&!5 #$*+,-67 (1) u ¯ = r(cos(−ϕ) + i sin(−ϕ))( (2) u · v = rs(cos(ϕ + ψ) + i sin(ϕ + ψ))( (3) u−1 = r−1 (cos(−ϕ) + i sin(−ϕ))( (4) u/v = r/s · (cos(ϕ − ψ) + i sin(ϕ − ψ))( 7" 0&12&134$5" "#$%&'( )!*$#" "+/#/3".) +&+"<*0 8'&D+)!/*&6+ &*0D+07A D#-G ,=-!90'00 v ∈ C "#$%&'( )!*$ ')'0B/ + z 7→ z + v &'"B%'!B) /'$ $*)A $3/0 + v FD'! 0+,0#!7 6'"0#,,+& 6+&7 '&0#&*) + "#$%&'( )!*$)9"#/1 "#$%&'( )!*$#" 0,3-#/#$'0,3".) +&+"<*0 8'&D+)!/*&6+ %'43- &*0D+07A D#-G ,=-!90'00 v = cos ψ + i sin ψ ')'0B/ + z 7→ z · v &'"B%'!B) /'$ $*)A $3/0 +! #,3-7 "=,:&3 ψ )!=-H 8#,-+0*) + "#$%&'( )!*$)9"#/1 8" 9$':;" ! 3)$',0 )!3/.)! B) "#)!3/.)! =))!'-!B)3 "B%&'0'"'0 "=//G'/ $'-"+%D+0<." "#$%&'( )!*$#" )'-90)B-B6'&1 K'"3/0):" + u = cos ϕ + i sin ϕ,

"#$%&'( )!*$#"+01

v = cos ψ + i sin ψ

0,3-#/#$'0,3".) +&+"#""+& )!*$#&6+ + )!#,!+0." u · v = cos(ϕ + ψ) + i sin(ϕ + ψ).

L' D+ + "+/#/3".) +&+"#0 D+)!/*&<." + )!#,!+0 "3)!*$#&*)*,+A +""#, u · v = (cos ϕ + i sin ϕ) · (cos ψ + i sin ψ) = cos ϕ · cos ψ + cos ϕ · i sin ψ + i sin ϕ · cos ψ + i sin ϕ · i sin ψ

= (cos ϕ · cos ψ − sin ϕ · sin ψ) + i(cos ϕ · sin ψ + sin ϕ · cos ψ).

236'& +! u · v "#$%&'( )!*$ '-GB,0'&$H'/ 9,D+07 8'& "+/#/3".) +&+"5+/A '!B,0 cos(ϕ + ψ) = cos ϕ · cos ψ − sin ϕ · sin ψ, B) sin(ϕ + ψ) = cos ϕ · sin ψ + sin ϕ · cos ψ.

M+)#/&7+/ )!*$90D+07 "3 + cos(ϕ − ψ) B) sin(ϕ − ψ) "B%&'0' 3)A 4' '""#, +! u B) v "#$%&'( )!*$#" D*/G+4#)*0 "'&& 6'//:/"1

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

$/$0)'

!"#$%& '$# ()"* z = r(cos ϕ + i sin ϕ) +,#-%$. /(!# )/ n ∈ Z z n = rn (cos(nϕ) + i sin(nϕ)).

2" 3$-4$5&/"

!"#$ %&' (&)&$ *+ &(,+, $"$&( -".(&$"$ )*/+%0(%! ."(102( * i0.123456 "#$"-"%&- 1&34 %!0(0/0)5+6 (2) 78*+4&9 )58: i−1 = −i6 (3) 78*+4& '!%1&% 0 6= z ∈ C "/ n ∈ N &/&$"%9 )58: |z n | = |z|n 6 (4) 78*+4& '!%1&% 0 6= z ∈ C "/ n ∈ N &/&$"%9 )58: z n = (z)n 6 (5) !(:&% ;5%*(5% )&(:&+-&1%&- &( * z ∈ C \ R ;*(<1! -5'.(&= /+0' &8"/+ )*$;0%:*!6 (6) &(:&- *+5- * z -5'.(&= /+0'5-9 *'&(:&-#& arg z = arg(z 2 )6 (7) &(:&- *+5- * z -5'.(&= /+0'5-9 *'&(:&-#& arg z = arg(z 3 )6 (8) &(:&- *+5- * z -5'.(&= /+0'5-9 *'&(:&-#& arg z = arg(z −1 )6 (9) &(:&- *+5- * z -5'.(&= /+0'5-9 *'&(:&-#& |z| = |z 2 |6 6" 7$'48" >21?2-9 )58: cos 2α = cos2 α − sin2 α "/ sin 2α = 2 sin α cos α@ &8'2$*$?2-9 )58: cos 3α "/ sin 3α )58:*% /+0'A$)*$< -! &8:/+&#B&%@ C&8:D- * z = cos α+i sin α -5'.(&= /+0'5$ "/ /+0'5(?2- -! * )*#'*1!- )*$;0%:0$ * $#!85%5'&$#!-2/ *(*-?* (1)

z 3 = cos 3α + i sin 3α,

"/ * -*%5%!-2/ *(*-?*! /&8A$/"8";&( EF&()*/+%0(;* *+$9 )58: (a + b)3 = a3 + 3a2b + 3ab2 + b3G z 3 = (cos α + i sin α)3 = cos3 α + 3i cos2 α · sin α − 3 cos α · sin2 α − i sin3 α

= (cos3 α − 3 cos α · sin2 α) + i(3 cos2 α · sin α − sin3 α).

>&)0$ *+$ -*.$2-9 )58:

cos 3α = cos3 α − 3 cos α · sin2 α, 2

"/

3

sin 3α = 3 cos α · sin α − sin α.

9" :&;<=>+?" >&$/+,(&8&/ n .5+!$A; &8"/+ /+0' "/ z ∈ C &/&$"% *+$ '5%1?2-9 )58: *+ u -5'.(&= /+0' n4&1!- 8:H-& z4%&-9 )* un = z@

12'3$' '$#'4%%5 z = r(cos ϕ + i sin ϕ) +,#-%$. /(!#'5+ -,'0,/5' n +6%7'87(9 ;&7+$ <5'= #);-$32;

@A" #$%&'" n:$32+

√ n

z=

√ n



ϕ + 2kπ ϕ + 2kπ + i sin r cos n n



(k = 0, . . . , n − 1).

@B" 7$'48" I+0'A$/2- -! *+ 1 -5'.(&= /+0'%*- * $!+&%-&$$&1!- 8:H-&!$9 "/ *1?2- '&8 ,-&$ -*%5%!-2/ *(*-J*%@ K+ 1 $#!85%5'&$#!-2/ *(*-?* $&#'"/+&$&/&% *+ 1 · (cos 0 + i sin 0)@ L&()*/+%0(;* * %&;&+&$&/ /+H8&- /+!%2/+0$ "/ -5/+!%2/+0$ *+$ -*.?2-9 )58: *+ 1 $!+&%-"$ 8:H-&M  0 0 + i sin = 1, u0 = 1 · cos 12 12   √ 2π 2π 3 1 u1 = 1 · cos = + i sin + i, 12 12 2 2 

u2 u3 u4 u5 u6 u7 u8 u9 u10 u11

√  3 4π 1 4π + i sin i = + = 1 · cos 12 12 2 2   6π 6π = 1 · cos + i sin = i, 12 12 √   8π 3 1 8π = 1 · cos =− + + i sin i 12 12 2 2 √   3 1 10π 10π = 1 · cos =− + i sin + i, 12 12 2 2   12π 12π = −1, + i sin = 1 · cos 12 12 √   3 1 14π 14π = 1 · cos + i sin − i, =− 12 12 2 2 √   3 1 16π 16π =− − + i sin i = 1 · cos 12 12 2 2   18π 18π = −i, = 1 · cos + i sin 12 12 √   3 20π 1 20π = 1 · cos + i sin i =+ − 12 12 2 2 √   3 1 22π 22π =+ + i sin − i, = 1 · cos 12 12 2 2 

! ε "#$%&'( )!*$#+ n,'-." '/0)1//02"3'" 3'4'!!5" 6n ∈ N+ 78 9: εn = 1; ! ε "#$%&'( )!*$ '/0)1//02"8 9: n,'-." '/0)1//02" 4:&:$'&0 n ∈ N+ ,<'; " +,-$." ! n"#$%& #'()*''(+&+& , &+-#.&#!/&0

!" #$%&'()*"

2kπ 2kπ + i sin (k = 0, . . . , n − 1). n n εk = εk1 4%5$#5 k = 0, . . . , n − 1 #)#.*56

εk = cos

1!!#2 , 3#2+2*))#2 ε0 = 1 *) /" 0$12$134,5" ! n,'-." '/0)1//02"2" '/0 )!:=*&0#) n,)!2/'+ :&"#+3:" : "#$%&'( )!*$, )>"#38 :$'&03'" : "2<5&><+ "2<' :! #<./? "2!1%%#3+@ '/0)1/"2<8 1) '/0." A)@A): 1; 6B! : "1+ .3C#<$*A.? '/01<+'&$D'3 $'/ .) 9:+*<#!!: :! n,)!2/'+;7 6" 7,.89" ! '&)E '/0)1//02"2" 9:&$:!: : { z ∈ C : z 1 = 1 } = {1}.

$*)#-." '/0)1//02"2" 9:&$:!: :

{ z ∈ C : z 2 = 1 } = {1, −1}.

9:<$:-." '/0)1//02"2" 9:&$:!: : 3

{z ∈ C : z = 1} =

3'/0'-." '/0)1//02"2" 9:&$:!: :

√ √ ) 1 1 3 3 i, − − i . 1, − + 2 2 2 2

(

{ z ∈ C : z 4 = 1 } = {1, i, −1, −i}.

9:+#-." '/0)1//02"2" 9:&$:!: : ( { z ∈ C : z6 = 1 } =

√ √ √ √ ) 1 1 1 3 3 3 1 3 i, − + i, −1, − − i, − i . 1, + 2 2 2 2 2 2 2 2

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

√ 3

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;6228.2 %..%%.2<,)!4.-<: 4$18 −2i 186(: #)=.% (−2i)3 = −8i3 = 8i/ >.4"- 4* *%(*%#*!!'( *! .%
," -&./0123" 0!- #$25&'(: 4$18 * ε ($#?%.A 3!"# ?,)#)-B= nC.5)( .183+1186(: 4* nC.5)( .183+1186(: 5. 2.# mC.5)( .183+1186( 3.##)%8.2 0 < m < n .1+3!,./ √

4" )$'*+" 0! 1 ?,)#)-B= .%3< .183+1186(/ 0 −1 ?,)#)-B= #"3$5)( .183+1186(/ 0 − 12 + 23 i √ 4*,#*5)( .183+1186(6(/ 0! i +3 −i ?,)#)-B= 2.18.5)( .183+1186(6(/ +3 − 21 −√ 23 i ?,)#)-B= √ 3 3 1 1 0! 2 + 2 i +3 2 − 2 i ?,)#)-B= 4*-$5)( .183+1186(6(/

2kπ >. εk = cos 2kπ n + i sin n $!"-4!!"0) ())*9 4- ?-() ())*9 +93%35:@ n1$23) $!"-4!!"0)8 6( k 9$'(5:@ +9:% n16$.; 57" #$%&'" > +93%35:@ n1$23) $!"-4!!"0)0) -./%( ϕ(n)8 (6*' ϕ (. &'$91A4'$ AB!!@4#";

56" #$%&'"

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$!"B556(5D- (an , . . . , a0 ∈ C) #$%)*#-5(#- (n ≥ 18 -./%*'@( +*#5*-(# n 2(9(E )*%+'$, !"0)$ @(#; 55" #$%&'" <$5-.F'$!$- z = a + bi )*%+'$, -./%9(

an 6= 0)

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z2 z3 z4 + + + · · · = ea · (cos b + i sin b). 2 3! 4! 5D" )$'*+ =EF'&:GHI:JF'+C" eiπ + 1 = 0/ ez = 1 + z +

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i(−1 − i) |1 − i|

11

,

  3+i 3−i (2 − i)(−3 + i) 1001 + , x= , (−1 + i)(3 − 2i) (−1 − i)(3 − 2i) |(2 − i)(−3 + i)| " √ " √
!389
79 7 3 − 21i (−2 − 2i) −5 3 + 15i √ , z= y= √
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w=

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?14! '() !

(−5 + 2i)x1 + (1 + i)x2 + (3 − 9i)x3 + 4ix4 = 1 − i (−3 + i)x1 + (1 + i)x2 + (2 − 7i)x3 + 3ix4 = 1 − i (1 − i)x1 + (1 + i)x2 + −3ix3 + ix4 = 1 − i

*,.(#$,0 ();(.*("$(.10&($ #*"!*#.%0 '()%*1#0#"7 !+! #$%&'&(! @>)(&&( (* !& !*#++, );A-B%.#0%-!"5 √ 15

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!"#$$ %& '()*+,-. !/.0!/.(+/ (((1 %2/+3,4/. 567781 !9,:(;) 6<1=

(5) n = 7, z = cos

!"! #$%&'&(!

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!"#$%&%'()"#*+, -.-*/-*0( - *12()*(34 *%'5.(6 ,37'%*8

 π π , t = 0 (cos π + i sin π) , u = 2 cos − i sin 4 4    π    π 3π 3π + i sin − + i sin − , w = cos . v = −1 cos − 3 3 4 4

9'(.:#* &(' )"#$%&%'()"#*+, -.-*/; -3) -<=- '($ )"#$%&%'()"#*+, -.-*>-&

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sgn(σ) · a1,1σ · a2,2σ · · · an,nσ .

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n = 3 !'!*>;B a11 a12 a13 a21 a22 a23 = sgn(id) · a11 a22 a33 + sgn((1 2)) · a12 a21 a33 + sgn((1 3)) · a13 a22 a31 a31 a32 a33

+ sgn((2 3)) · a11 a23 a32 + sgn((1 2 3)) · a12 a23 a31 + sgn((1 3 2)) · a13 a21 a32 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a12 a21 a33 − a13 a22 a31 − a11 a23 a32 ,

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dim(U ∩ V ) + dim(U + V ) = dim(U ) + dim(V ).

! R4 6/+(*7(27%/8 ./1"9<=+ "! (1, 1, 2, −1)5 (−2, 1, 0, 1) 2) (−3, 3, 2, 1) 6/+> (*7*+ $#("# 1/8/7$#( "#(/7/( /#/./&8/+ ./1"9$)$6"#; ?=9<=+5 3*10 " 1/8/7$#( U "#(27 /!/8 6/+(*7*+ #&8/$7&) +*.%&8$@&A<"5 "!"! 5" 6$,&7"

U = { x1 · (1, 1, 2, −1) + x2 · (−2, 1, 0, 1) + x3 · (−3, 3, 2, 1) : x1 , x2 , x3 ∈ R }.

?/3$( " (b1 , b2 , b3 , b4 ) ∈ R4 6/+(*7 "++*7 2) @)"+ "++*7 /#/./ U >8"+5 3" "!      b1 1 −2 −3 x1 b2   1 1 3  · x2  =    b3  2 0 2 x3 b4 −1 1 1 

#&8/$7&) /10/8#/(7/89)!/78/+ 6"8 ./1*#9$)" "! x = (x1 , x2 , x3 ) 6/+(*77" 82!6/; B/#'7<=+ "! /10/8#/(7/89)!/7 %C6'(/(( .$(7&D$(5 2) " E"=))>/#&.&8$@&A )/1'()2126/# ./1*#9<=+F 

  1 1 −2 −3 b1  0 1 b 1 3 2  ∼ 2 0 2 b3  0 0 −1 1 1 b4

−2 3 4 −1

  b1 −3  b2 − b1  6 ∼   8 b3 − 2b1 −2 b4 + b1

1 0 0 0

−2 0 0 −1

 −3 b1 0 b2 − b1 + 3(b4 + b1 )   0 b3 − 2b1 + 4(b4 + b1 ) −2 b4 + b1

G/. 621/!(-+ /# " E"=))>/#&.&8$@&A .&89/8 #2H2)2(5 ./7( " %/+/7/(/!/(( /#/./+/( ("7("#."> !A )*7*+"( 8/. )!*7*!(=+ %/ /10 *#0"8 +*8)("8))"#5 3*10 " %/+/7/(/!/(( /#/./+ .&89/10&+/ 1 #/10/85 .$)72)!( )*7@)/72++/# 8/. 3*!(=+ " .$(7&D*( #2H@)C) "#"+7"; +"H*(( %C6'(/(( .$(7&D7A# .$7 #$(<=+5 3*10 x1 2) x2 +,(,(( 6$#(*!A+ I./7( "! /#)C 2) .$)*9&+ *)!#*H%"8 6"88"+ " +&(-8(/(/(( /#/./&8+J5 2) "! /10/8#/(7/89)!/7 "++*7 2) @)"+ "++*7 *#93"(A ./15 3" b2 = b1 − 3(b4 + b1 ) = −2b1 − 3b4 , 2) b3 = 2b1 − 4(b4 + b1 ) = −2b1 − 4b4 .

?/3$( "!( .=("((=+ ./15 3*10

U = { (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) : b1 , b4 ∈ R }.

K##/8C7!2)+2HH ./182!!-+5 3*10 " 1/8/7$#A 6/+(*7*+ (280#/1 /%%/8 "! "#(27%/8 6"88"+; ! (1, 1, 2, −1) 6/+(*7 /)/(28 b1 = 1 2) b4 = −15 (/3$( −2b1 −3b4 = −2·1−3·(−1) = −2+3 = 1 2) −2b1 − 4b4 = −2 · 1 − 4 · (−1) = −2 + 4 = 25 ".& 2HH/8 " 3/#0/) 27(2+; L")*8#A"8 (−2, 1, 0, 1)>7/ −2 · (−2) − 3 · 1 = 4 − 3 = 1 2) −2 · (−2) − 4 · 1 = 4 − 4 = 0; M21/!/(-# " (−3, 3, 2, 1) 6/+(*77" −2 · (−3) − 3 · 1 = 6 − 3 = 3 2) −2 · (−3) − 4 · 1 = 6 − 4 = 2; 8" 6$,&7" ! /#C%%& H2#9" U "#(2728/+ "! U = { (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) : b1 , b4 ∈ R }.

!"#$%%$# &'#( )$*'+,-,.(# /'*01/ -1 2'-3/1- 4/516),74(6' 8&$% $3%$%2$%9/ : ;#<-386 4-= U !%+4)$/34(- )$6% b1 !- b4 -3'.'+1/ &,#'-3%2'%(= !- $*0 .,34-,% >*0 /0$62$%?9 = 21*0 ' b1 = 1, b4 = 0 $-$%!/ (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) = (1, −2, −2, 0), !-

b1 = 0, b4 = 1 $-$%!/ (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) = (0, −3, −4, 1),

'3'3 '3 (1, −2, −2, 0)= (0, −3, −4, 1) &$ %166$/+-3$6 .,34-' U @/' : A3 4- 8//0$/ #,%2'@ %(= 21*0 ' (0, 1, 0, 0) !- (0, 0, 1, 0) &$ %161 /4/7-$/$ '3 U '#%!6.$/= !- 1#0'/ W '#%$6$% *$/$6,#/' = ')$#06$ U ∩ W = {0} !- U + W = R4 B$*0 4#0$/ W '#%$6$% '3 U 1)"#$)$/@ %C),/' 2D&?C E: F$*09 !-36$= 21*0 %$#?$-9# '3 '#%$6$ +4)$/34(%!%$#$= )$6% dim(U ) = 2= dim(W ) = 2= dim(U ∩ W ) = 0 !- dim(U + W ) = 4:

!" #$%&'" G$ 4/%-9 ' (0, 3, 1, −1) !- (1, −5, −9, 1) &$ %161 ,#%'# *$/$6,#% V '#%$6$% '3 R4 &$ %16%!6.$/= !- '+?C )$* ' HI: "!#+,213 2'-1/#('/ ' V $#$)$4% !"#$% -$*D%-!*!&$#: A )$*5$#$#< $*0$/#$%6$/+-3$6 .<&D%$%% ),%64J' !- '//' K'C--@5!#$ $#4)4/,74(&'# &'#( ,%'#' D@ %,-'L       0 1 c1 1 c1 0 c1 0 1  3 −5 c2  0 22 c2 − 3c3   0 0 c2 − 3c3 − 22c1   .   ∼  1 −9 c3  ∼ 1 −9    c3 1 −9 c3 −1 1 c4 0 −8 c4 + c3 0 0 c4 + c3 + 8c1 A3% '"%C %$2,%= 21*0 '3 $*0$/#$%6$/+-3$6/$ ' 16 !- 7-' ' 16 &'/ )$*1#+,-'= 2' c2 − 3c3 − 22c1 = 0 !- c4 + c3 + 8c1 = 0: G$2,% ' V '#%!6 $#$)$4/$ !"#$%%$# &'#( )$*'+,-' V = { (c1 , 3c3 + 22c1 , c3 , −c3 − 8c1 ) : c1 , c3 ∈ R }.

(" #$%&'" A HI: !- HM: "!#+, '% '3!6% 1#+1%%C )$*= 21*0 '3 1%% 4-3,)D%1%% $6$+)!/0$ 5$#2'-3/,#,-,&'# '3 U !- V '#%$6$ )$%-3$%!% 4-3,)D%-C : ;++4* %C+?C = 21*0 U = { (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) : b1 , b4 ∈ R }, !-

V = { (c1 , 3c3 + 22c1 , c3 , −c3 − 8c1 ) : c1 , c3 ∈ R }.

G$2,% 2' &$-39/ $*0 838- &$ %16% ' !% '#%!6.<#= ' 16 '3'3

(b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) = (c1 , 3c3 + 22c1 , c3 , −c3 − 8c1 ), b1 −2b1 − 3b4 −2b1 − 4b4 b4

= c1 , = 3c3 + 22c1 , = c3 , = −c3 − 8c1 .

;3 )$*4/% 7-' $*0 B21)1*!/E #4/$,64- $*0$/#$%6$/+-3$6= ')4% -1 5!#$ )(+-3$66$# )$* %C+C/ 1#+'/4= "#: K'C--@$#4)4/,74(&'#: N1-% 7-' ' ?(# .$&,#% 83!"4- 1#,- )(+-3$6% '# '#)'33C L c1 !- c3 @'% 45$?$339 b1 !- b4 -$*D%-!*!&$#= !- '"?C = 21*0 −2b1 − 3b4 = 3(−2b1 − 4b4 ) + 22b1 = 16b1 − 12b4 ,

O%6$/+$339 L

b4 = −(−2b1 − 4b4 ) − 8b1 = −6b1 + 4b4 . 9b4 = 18b1 , −3b4 = −6b1 ,

')$#0/$ )$*1#+,-' !""$/ b4 = 2b1 = '21# b1 %$%-3<#$*$- -3,) #$2$%: A3% '"%C %$2,%= 21*0 U ∩ V = { (b1 , −2b1 − 3b4 , −2b1 − 4b4 , b4 ) : b1 ∈ R !- b4 = 2b1 } = { (b1 , −8b1 , −10b1 , 2b1 ) : b1 ∈ R }.

P*0 U ∩ V $*0+4)$/34(- '#%!6 !- '3 (1, −8, −10, 2) &$ %16 *$/$6,#?':

!" #$%&'" !"#$%$&'( % )*+ ,- ).+ /,"01(2%3 245464$4$$ U ,- V %"$474( 586-91"%$1$+ :0089

$'0&'(; $,$4",2?" (%/&'(;
@6 4"?6? =>0-6474((4" =17 =49 8- $'0&'( (474-38 U + V 2168-1$+ A'0&'(; &% 943471"&%; %6%6 U + V = [(1, −2, −2, 0), (0, −3, −4, 1), (0, 3, 1, −1), (1, −5, −9, 1)].

:334( %6 %"$,734( %6 4"4=48$ =4983$ (,/"4$ -49F$-,9,54" %0&'( =49; %=82?" % 2168- =17 (8!"B 5%-<%$>+ @ =49G4"4"? "834178- 49#43"4$7430-647 2?5F$4$$ =1$78H% ,- %33%( I%'--B4"8=831J8>5%" 5%"> =49!"01-%K 

   d1 1 0 0 1 1 0 0 1 d1 −2 −3 3 −5 d2  0 −3 3 −3 d2 + 2d1      −2 −4 1 −9 d3  ∼ 0 −4 1 −7 d3 + 2d1  0 1 −1 1 d4 0 1 −1 1 d4  1 0 0 0 0 0 ∼ 0 −3 0 1 −1 0

 1 d1 0 d2 + 2d1 + 3d4   −3 d3 + 2d1 + 4d4  1 d4

@6%6; %6 49#43"4$7430-64734( %((!7 ,- J-%( %((!7 5%3 =49!"01-%; <% d2 + 2d1 + 3d4 $4<1$

= 0;

U + V = { (d1 , −2d1 − 3d4 , d3 , d4 ) : d1 , d3 , d4 ∈ R }. :22?" =17 (L33#43 "4 $'0&'( !"5%-38; - 54($!7$,7; ,- (1, −2, 0, 0); (0, 0, 1, 0); (0, −3, 0, 1) 49# 2168-%+

(" )*+,-./0" @ T $4-$ G4"4$$8 V 54($!7$,7 v1 , . . . , vk 54($!77430-647 "834178-%3 GD994$"43

7,-67430-6474834( =%H8=1"8-

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