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/"#**!* *)-:#'+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 yRx′T y′. EK4LF
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!"#$%$&# '()(* ! "#$!%&'( R #)% *&%# !"#$%&'() * +,!&%) +"#),#-%&.#/#(% * -$. /)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 yRx′T y !+x′T yRx′T y′⇒xT yRx′T y′4 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 y−1∈H⇔y∈Hx
8"#4.# ; M- R !*+ /&=(#+-."! 2 ,&-+!> xRy ⇔ xT y−1Re⇔ xT y−1 ∈cl(e) !+ &' :<,-7! @)! H = cl(e) !*+
.-!' )' *&)*1C,&)(! !G4 O</-(,&@)!=!'+ *- &' #2.>xT y−1∈H⇔(xT z)T(yT z)−1 ∈H>R!*+ /&=(#+-."!
2 ,&-+!4
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+,!&-./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 y−1 ∈H ⇔x−1T y∈H7
!"# $!%$&"#' !% (!%%' &)
"#$%&'&(% )* *+ !"# H ⊳ G$ %& '()&#"!* R+,-*"( .&'
xRy⇔xT y−1 ∈H
(/# &..(),( !"#$%&' ()*+,%-#"!'.! /0"1&*# )( /!0/23'!0.( H$ %4(*/(56)( 70!#"(*# G/R(/# &)!'/ 0* 3'!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.
✄
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f : (x, y)→x+y
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!%, ('/ +;)9'/ (+/,+%$,'/5
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!"# $%&'(#&)*+,- #( (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*% ,.",*# !*# $%.$%(&,&# !(&*# : '*,,*
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#%) G′7 86')# f−1 "#$ %! +#'(')*&+#(" 2"G′ #%) G0
9)"%:" 7 ?( f *#, "+ (#.).%$5(#)*- 4!.%#f *#, "+* =(>*',(.+- /.+'f−1 4"##(; <! #"@, /* ).+,%*% 6"*f−1 *#,
"+ ).%$5(#)* /* 1%."$*#; ?.(*+,x *,y /*"0 &!&)*+,# 6"*!'.+6"*# /*G′; A+ 4 4!.%# 7f(f−1(x)T f−1(y)) = f(f−1(x))⋆f(f−1(y)) =x⋆y; B2.Cf−1(x)T f−1(y) =f−1(x⋆y)*,f−1*#, /.+' "+ (#.).%$5(#)* /* 1%."$*#
/* G′ #"%G;
"#$%&'&(% )* *+) ;')#5%" G=G′7 %! &'('(')*&+#(" "#$ 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.).).%$5(#)*- '4%
f(x+y) =e2iπ(x+y)=e2iπx×e2iπy =f(x)×f(y)-∀x, y∈R;
,-(.(/&'&(% )* *++ ;')#5%" G *'##>2" %! <6<("!$ !"%$)" e7 6=+(84" &'('(')*&" f(e) = n "#$ !"%$)"
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2" x 28!# G0