GHGʬȤ롣ΤȤgH{gh;hH}ν礽ΤΤǤꡢgޤHκ;ࡢHg={hg;hH}gޤHα;Ǥ롣;νG/H;νH\Gɽ
ΤȤ黻ɽμˤʤȤΤʤФʤʤ
ŪˤȡΩĤȤǤ롣
aH=a'H,bH=b'HעΡ(ab)H=(a'b')H
줬ΩĤˤϡǤդa,bG;h,h'HФơ(ah)(bh')=(ab)hhH¸ߤ뤳ȡפɬʬǤ롣
δطơʬ롣
[]HϷGʬǤȤ롣
ȤˡʬHGʬnormal subgroupȸƤ֡
[]ʬξϼΤ褦Ѥ뤳ȤǤ롣ƱͤǤ롣
[]
[1](1)(2)
ƱƱΩġ
[2](2)(3)
alb (mod H)
arb (mod H)
acrbc (mod H)
aclbc (mod H)(*)
ޤƱͤˤcld (mod H)bclbd (mod H)(**)
(*)(**)ȿΧꡢaclbd (mod H)Ωġ
[3](3)(4)
baHaH=bHalb (mod H)ꡢ(3)̿ϼΤ褦˽롣baH,dcHabacHǤ롣Ĥޤꡢa,cG;aHcHacHǤ롣
c=a-1ȡ
aHcHacH
aHa-1Haa-1H=eH=H
aHa-1H
ޤaHa-1aHa-1HʤΤǡaHa-1H
[4](4)(5)
(4)aǤդʤΤǡa-1ǹͤȡa-1HaHΩġ
HǤդθhȤȡa-1haHǤ뤫顢HΤ븵h'äơa-1ha=h'Ȥʤ롣äơh=ah'a-1aHa-1Ȥʤ뤫顢HaHa-1Ǥ롣
ޤ(4)Ȥ߹碌ơaHa-1=HΩġ
[5](5)(1)
aHǤդθah(hH)ɽΤǡ
aha-1
aHa-1
=Hʢ貾(5)
äơaha-1=h'HΩġ
ah=h'aHaȤʤΤǡaHHaΩġƱͤˤHaaH⼨롣
äơaH=HaΩġ
[]GĴʤ顢٤ƤʬʬǤ롣
[]GĴΤȤϡaG;aH=HaΩġ
椨ˡĴʬϤ٤ʬǤ롣
[](Z,+),(Zn,+),(Q*,),(RSUP{*},),(C*,),(Mn(K),+)ʤɤϤ٤ƲĴǤΤǡʬϤ٤ʬǤ롣
㤨СZʬmZʬǤ롣
[]CnüSLn(C)ϡnGLn(C)ʬǤ롣
ʬפǤͳϡʬHξ;ƱΤ黻뤳ȤǤȤǤ롣
[]
[]GˤơG濴Z(G)ʬǤ롣
[]GʬHλؿ2ʬʬǤ롣
¨[G:H]=2Hʬ
[]G=H+gH=H+HgϡGα;Ⱥ;ؤʬǤ롣
ʬ϶̸ʤΤǡgH=HgǤʤФʤʤäơHʬǤ롣
Ϥ褯Ȥ롣
[]νƱաGHγKer()GʬǤ롣
[]Ǥդ˦ǢKer(),aGȤȤ
(aa-1)
=(a)()(a-1)ʢդϽƱǤ뤫
=(a)(a-1)ʢKer
=1
äơaa-1Ker()Ωġ
[]ʬˤʬͭʬȸäƤ롣ȯ뤳Ȥˤꡢ
[]GμʬϤɤGʬǤ롣
[]GμʷȤGȤȡñ̷{e}ΤȤǤ롣
[1]N=GΤȤ
g-1ngG=NǤ뤿ᡢGGʬˤʤ롣
[2]N={e}ΤȤ
g-1ng
=g-1egʢNθeʤᡢn=e
=g-1g
=e
N
äơNGʬˤʤ롣
[]оηS3ˤơʬN=<(1 2 3)>={(1), (1 2 3), (1 3 2)}ʬˤʤ롣
N
=<(1 2 3)>
=<d>ʢ跲ɽ
={d,d2,d3}
={e,d,f}ʢ跲ɽ顢d2=fd3=e
ϡn{e,d,f}(=N)g{a,b,c,e,d,f}(=G=S3)줾ǡg-1ngNǤ뤳Ȥǧ롣
ĤΡn{d,f}g{a,b,c}Ȥ߹碌6ѥǡg-1ngNǤ뤳Ȥǧ롣
ʾꡢɤξǤg-1ngNΩĤᡢNS3ʬǤ롣
<(1 2)>={(1), (1 2)}S3ʬˤʤ뤬ʬˤϤʤʤ
ʤʤС<(1 2)>={e,b}ǤꡢΤ褦˷Ǥ롣
a-1ba
=abaʢ跲ɽ顢a-1=a
=da
=c
c<(1 2)>°ʤǤ롣
GʬHǾ;ʬ䤹Ȥ;ȤΤȱ;ȤΤȤǤϷ̤ۤʤ뤳Ȥ롣HʬΤȤϡ;ȱ;बפ롣ĤޤꡢɤΥ饹ʬȤäƤ⡢Ʊ饹ʬˤʤ롣
[]NGʬʤСGǤդθaФơaN=NaΩġ
嵭μκդ;ࡢդ;Ǥ롣
[]
[1]aNNa
aNθan (nN)ȤɽȤǤΤǡΩġ
an
=ana-1a(e=a-1a)
=(a-1)-1na-1a
Na(NGʬǡa-1GnNǤꡢ(a-1)-1nNǤ뤫)
äơaNθϤ٤Na°롣
[2]NaaN
Naθna (nN)ȤɽȤǤΤǡΩġ
na
=aaSUP{-1}na(e=aa-1)
aN(NGʬǤ뤿ᡢa-1naNΩĤ)
äơNaθϤ٤aN°롣
äơ[1][2]ꡢaN=NaΩġ
ʬǤϺ;ȱ;बפ뤿ᡢñ;Ȥ
[]G1,G2H1,H2줾ʬȤ롣
ʬǤꡢΩġ
[]ͽƱȤ߹碌ơͽƱСγˤ
Ǥ롣
[]ñ㷲
Gʬ{e}GʳʬʤȤGñ㷲Ȥ
ʬʳΥåȤϤ켫ΤǤϷǤϤʤ
[]Gʬä硢GHʬåȤϷˤʤ롣
[]HʬǤȤϡGΤ٤ƤθФơH=HΩĤȤ̣롣
GʬG=H+H+H+H+ĤȤȡΩġ
HH
=HHʢH=H
=H
Ȧ¤GθʤΤǡ¤GθǤ롣ĤޤꡢHϥåȤΤɤ줫ˤʤ롣
;ñ̸HǤ롣ޤHεոϦ-1HȤʤ롣
äơ
H-1H
=-1HH
=H
=H
NGʬȤȤGNˤ;11Ĥȹͤơʽªˡ;Τκ뽸G/NȤɽ
[]GNˤäơ3ĤΥ饹ʾ;N,aN,bNʬ줿Ȥ롣
ΤȤ
G/N={N,aN,bN}
Ωġ
ĤޤꡢG/NϽνǤ롣
G/N˼ˡѤ롣
(aN)(bN)=abN
aN=a'NbN=b'Nע͡abN=a'b'NפΤƤɬפ롣
[]NGʬΤȤG/N(aN)(bN)=abNѤСG/NϷˤʤ롣