ΥڡϤƤʥ֥åޡɲΥڡޤϤƤʥ֥åޡ Υڡlivedoor åפɲΥڡޤlivedoor å

ܼ

1-1򸵤Ȥ뽸{1,-1}ͤ롣Ѥ˴ؤƷɽ񤯤ȼΤ褦ˤʤꡢˤʤ뤳Ȥ狼롣

1-1
11-1
-1-11

Ĥޤꡢ{1,-1}ϳݤ˴ؤĤƤơ̾ογݤʤΤ餫˷ˡ§Ωġηñ̸1-1εո-1Ǥ롣

nоηSn鷲{1,-1}ؤμf򼡤Τ褦롣

  • xִΤȤ¨ĤθߴѤɽִΤȤ
    • f(x)=1
  • xִΤȤ¨ĤθߴѤɽִΤȤ
    • f(x)=-1

ȡf:Sn{1,-1}Ʊˤʤ롣
ʤʤСߴθĿͤȡΤ褦ˡf(xy)=f(x)f(y)פΩĤǤ롣

  • (ִ)(ִ)ִ
  • (ִ)(ִ)ִ
  • (ִ)(ִ)ִ
  • (ִ)(ִ)ִ

äơ[]f:G1G2G1鷲G2ؤνƱȤ롣ΤȤfγKer(f)ϡG1ʬǤפꡢƱf:Sn{1,-1}פγKer(f)SSUB{n}ʬˤʤ롣

ǡKer(f)Anɽnθ已Ȥ'A'"alternating group"ƬʸǤ롣

Ĥޤꡢ已AnоηSnʬˤʤ櫓Ǥ롣

[]nθ已Anɽ

An
=Ker(f)
={xSn|f(x)=1}ʢ1{1,-1}ñ̸Ǥ뤫
={1,2,,n}ζִΤ뽸硡ʢf(x)=1ȤȤϡxִǤפȤȤƱ

已θ

[]3θ已A3θ򤹤٤Ƶ衣

[]3оηS3θ6(=3!)ĤǤ롣6ĤϼִǤʤ٤ƽִˡ

  • (1)
  • (2 3)
  • (1 2)
  • (1 3)
  • (1 2 3)=(1 2)(1 3)
  • (1 3 2)=(1 3)(1 2)

椫ִ¨ĤθߴѤõȡ3ĤǤ롣

  • (1)
  • (1 2 3)=(1 2)(1 3)
  • (1 3 2)=(1 3)(1 2)

3ĤθA3θǤ롣

已ִδط

[]已ִʬǤ롣

已ȸ已δط

[]已AоηS
֢פϡִޤפȤ̣εǤ롣

已ΰ̿

[]n1ΤȤ已Anΰ̿n!/2Ǥ롣
¨Ωġ

|A_n|~=~\frac{n!}{2}

[]f:Sn{1,-1}ФơƱŬѤ롣

S_n~/~\mathbb{Ker}~f~\conq~f(S_n)

n1顢ߴ(1 2)Sn°Τǡf*1=-1ꡢf(Sn)={1,-1}Ȥʤ롣

äơS_n~/~A_n~\conq~\{1,-1~\}Ωġ

ǡΩġ

|Sn|
=(Sn:An)|An|ʢ饰󥸥
=|Sn/An||An|ʢ辦ΰ̿ϡ;θĿǤ뤿ᡢ|Sn/An|=(Sn:An)
=2|An|ʢ[]Ʊʷΰ̿פꡢ|Sn/An|=|{1,-1}|=2

äơΩġ

|An|
=|Sn|/2ʢ嵭η̡
=n!/2ʢSnΰ̿n!Ǥ뤫ˡ

ʸ

  • طʤ󤫤勞ʤ
  • 13Ф̼˸륬οء


*1 1 2