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Ϥʬǡ켫ȤޤƱ¤򤷤ƤΤʬϤȤ

[]Xʬ礬ȤϤƱ黻˴ؤƱƤȤʬXȸƤ֡
XġΤΤȤʬʬġʬΤȤ
ޤX֤ΤȤʬ֤Ȥʤ롣

[]ʬȡ˴ؤƺ;ν礬ͤ롣

ʬ

ִ

оηSnʬִ̤ȤäˡִΤνAnʬȤ

ʬ

GθgǤդӡΤ٤Τͤ롣GʬgˤGʬ󷲤ȸƤ֡ gn=eȤʤn¸ߤСͭ½󷲤ˤʤ롣ǤʤСˡZƱʲĴȤʤ뤳ȤΤƤ롣

ü

GL(n,R)Τǡ󼰤1ΤΤ򽸤ʬ''ü''ȤSL(n,R)ɽ ʬȤϡ󼰤ˡŪǤ뤳Ȥ狼롣

ľ

ľO(n)ϡGL(n,R)ʬˤʤ롣ľȤ ˡľ򷲤Τǹ󼰤ͤ1ΤΤϡľ򷲤ʬüľȤSO(n)ɽ

˥⥸顼

üSL(n,R)ΤǡǤΤΤϡʬ˥⥸顼ȤSL(n,Z)ɽ Ǥιι󼰤1ʤ顢չǤˤʤ롣ϡ;ҹˤչɽ狼롣

˥귲

ǤդK˳ĥơGL(n,K)Ƚ񤯡ϡKθǤȤĵդʹΤȤǤ롣 ΤȤK=CʣǿΡˤȤȡGL(n,C)ϥ˥^t~\bar{U}=~UUˤΤǤꡢ˥귲ȤU(n)ɽ롣

ʬĤ

ͭĤζ

ͭZǶΤϡʬĤˡñ̸ϴޤޤʤ

ͭΤʬ

ͭQĤȤߤʤȤZʬĤȤʤ롣 ޤʬ줬ǿpΤ٤ͭʬĤ

·Ĥʬ

·M(n,R)ǤȤM(n,Z)ʬĤˤʤ롣

ʬΤ

ʣǿΤʬ

¿ΤϡʣǿΤʬΤǤ롣 ơͭΤϡ¿ΤʬΤǤ롣 ĤޤꡢͭΤϡʣǿΤʬΤǤ⤢롣

·1ѿͭؿΤʬ

·1ѿͭؿR(x)Ǽ¿ΤʬΤR(x)ΤȤ

ʣǷ1ѿͭؿΤʬ

·1ѿͭؿR(x)ϡʣǷ1ѿͭؿC(x)ʬΤ

ʬϤȽ

Xʬ礬ʬϤˤʤ뤿ˤϡ黻ĤƤ뤳Ȥɬ׽ʬǤ롣ĤޤꡢϤɬפʱ黻η̤ʬ礫æƤʤȤ׵ᤵ롣

ʬȽ

[̿]GʬHʬˤʤˤϡξ郎ΩĤȤɬ׽ʬǤ롣
(1)HGñ̸eޤࡣ
(2)x,yHʤ顢x~\circ~y~\in~H
(3)xHʤ顢x-1H

3Ĥξ狼򸺤餷ơ1ĤξȽǤ롣

[̿]GʬHʬˤʤˤϡξ郎ΩĤȤɬ׽ʬǤ롣 (4)\forall~x,y~\in~HФơx~\circ~y^{-1}~\in~H

[][1](1)+(2)+(3)(4)פ򼨤 yH(3)顢y-1H xy-1Фơ(2)ŬѤȡx~\circ~y^{-1}~\in~H

[2](4)(1)+(2)+(3)פ򼨤 xHꡢy=xȤ롣 (4)ŬѤȡx~\circ~y^{-1}~=x~\circ~x^{-1}~=e\in~H졢(1)Ωġ

yHꡢx=eȤ롣 (4)ŬѤȡx~\circ~y^{-1}~=e~\circ~y^{-1}~=y^{-1}\in~H졢(3)Ωġ

(4)yy-1ŬѤȡ x~\circ~(y^{-1})^{-1}~=x~\circ~y~\in~H졢(2)Ωġ

ʬĤȽ

[̿]AζǤʤʬBʬĤȤʤ뤿ˤϡξ郎ΩĤȤɬ׽ʬǤ롣
(1)\forall~x,y~\in~BФơx-y~\in~B
(2)\forall~x,y~\in~BФơxy~\in~B

ʬΤȽ

[̿]K2İʾθޤʬLʬΤȤʤ뤿ˤϡξ郎ΩĤȤɬ׽ʬǤ롣
(1)\forall~x,y~\in~LФơx-y~\in~L
(2)\forall~x,y~\in~L~\setminus~\{0\}Фơx/y~\in~L

ʬ

[̿]
(1)HSUB {1};H2G2ĤʬʤСH_1~\cap~H_2ʬˤʤ롣
(2)HSUB {1};H2G2ĤʬʤСH_1~\cup~H_2ʬˤʤȸ¤ʤ
(3)HSUB {1};H2G2ĤʬʤСH_1~H_2~:=\{~h_1~\circ~h_2~|~h_1~\in~H_1,~h_2~\in~H_2~\}ʬˤʤȸ¤ʤ

[]x,y~\in~H_1~\cap~H_2Ȥ롣
ΤȤx,y~\in~H_1Ǥ롣
ʬϱ黻ĤƤΤǡx~\circ~y~\in~H_1
ޤx,y~\in~H_2Ǥ롣
ʬϱ黻ĤƤΤǡx~\circ~y~\in~H_2
äơx~\circ~y~\in~H_1~\cap~H_2
ʬȽ(2)

ޤñ̸HSUB {1};H2ΤɤˤޤޤΤǡH_1~\cap~H_2ˤޤޤ롣
ʬȽ(1)

xεոˤĤƹͤ롣
H1GʬʤΤǡx~\in~H_1x^{-1}~\in~H_1Ǥ롣
Ʊͤˡx^{-1}~\in~H_2Ǥ롣
äơx^{-1}~\in~H_1~\cap~H_2
ʬȽ(3)

äơH_1~\cap~H_2ʬˤʤ롣

(2)H_1~\cup~H_2ʬˤʤפȤȿ򼨤

g_1~\in~H_1,~g_2~\in~H_2Ǥ뤬g_1~\circ~g_2~\not{\in}~H_1~\cup~H_2ȤʤΤõФ褤

㤨СH_1~=~\{e,~(1,2)~\}~\subset~S_3,~H_2~=~\{e,~(1,3)~\}~\subset~S_3Ȥ롣
g_1~\circ~g_2~=(1,2)~\circ~(1,3)~=~(1,2)(1,3)=(1,3,2)~\not{\in}~H_1~\cup~H_2ȤʤꡢȿƤϤޤ롣

(3)H_1~H_2~:=\{~h_1~~\circ~h_2~|~h_1~\in~H_1,~h_2~\in~H_2~\}ʬˤʤפȤȿ򼨤

H_1~H_2~\supset~H_1~\cup~H_2Ǥ뤳Ȥθơ h_1~\in~H_1,~h_2~\in~H_2Ǥ뤬h_1~~\circ~h_2~\not{\in}~H_1~H_2ȤʤΤõФ褤

㤨С(2)ȿǾҲ𤷤H1,H2ˤĤơH1H2ͤȡΤ褦ˤʤ롣
H_1~H_2~=~\{~e,~(1,2),~(1,3),~(1,3,2)~\}
νϡ(1,3)(1,2)=(1,2,3)ޤޤʤΤǡʬˤʤʤ׳ǧ

[]嵭̿(3)ˤĤơH_1~H_2~:=\{~h_1~h_2~|~h_1~\in~H_1,~h_2~\in~H_2~\}ʬˤʤפΩĤˤϡH1,H2ΰʬǤФ褤

ʬĤ

[̿]
(1)BSUB {1};B2A2ĤʬĤʤСB_1~\cap~B_2ʬĤˤʤ롣
(2)BSUB {1};B2A2ĤʬĤʤСB_1~+~B_2~:=\{~b_1~+~b_2~|~b_1~\in~B_1,~b_2~\in~B_2~\}ʬĤˤʤȸ¤ʤ

[](1)

[1]¤ˤĤ
B_1~\cap~B_2ʬ÷ˤʤ뤳Ȥϡ[̿]HSUB {1};H2G2ĤʬʤСH_1~\cap~H_2ʬˤʤפ̤ʥʤΤΩġ

[2]ѤˤĤ
[̿]HSUB {1};H2G2ĤʬʤСH_1~\cap~H_2ʬˤʤפξƱͤˤФ褤ʱ黻Ѥǹͤˡ

׳ǧ

(2)B_1~+~B_2ʬ÷ˤʤ뤬ѤˤĤɬĤƤʤ

㤨СA=\{~a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}~|~a,b,c,d~\in~\mathbb{Z}\}ϴĤǤ롣
ޤB_1~=~\{~a+b\sqrt{2}~|~a,b~\in~\mathbb{Z}~\},~B_2~=~\{~a+b\sqrt{3}~|~a,b~\in~\mathbb{Z}~\}ʬ֤Ǥ롣
B_1~+~B_2=\{~a+b\sqrt{2}+c\sqrt{3}~|~a,b,c~\in~\mathbb{Z}\}ȤʤꡢʬĤǤʤ

ʸ

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