ܼ

[]GθaФơanηθ(nΤκ뽸<a>ɽ

a>={, a-3, a-2, a-1, a0, a1, a2, a3, }

[]G=<a>Ȥ<a>ǤȤaϤȤ

1ĤȤϸ¤ʤ

[]Zϲˡ˴ؤ̵½󷲤ǡ1,-1Ǥ롣

ʬ

[]<a>GʬǤ롣

[]<a>Gθ뽸Ǥ뤿ᡢʬȽˡѤ롣

[]

[1]ޤ<a>ѤĤƤ뤳Ȥ򼨤

a>°븵a߾Ǥ뤿ᡢ2Ĥaman=am+nȤʤꡢa߾Ǥ롣

äơ<a>ѤˤĤĤƤ롣

[2]a0ñ̸顢ñ̸<a>°롣

[3](an)-1=a-n顢<a>°븵εոϡޤ<a>°롣

椨ˡ[1][3]ꡢ<a>GʬǤ롣

[]Gʬ<a>ηΤΤGʬȤ

[]GȤʬȤ¨G=<a>ǤȤGȤ
ΤȤaϽGȤ

󷲤ȥ٥뷲δط

[]󷲤ϥ٥뷲Ǥ롣

[]󷲤ϷǤ뤿ᡢˡ§ǧФ褤

[]G=<a>󷲤Ȥ롣

2Ĥθamanȡ

  • aman=am+n
  • anam=an+m

ǡm+n=n+nǤ뤳ȤѤȡaman=anamΩġĤޤꡢˡ§Ωġ

椨ˡGϥ٥뷲Ǥ롣

󷲤ΰ̿

[]Gθaΰ̿ϡʬ<a>ΰ̿˰פ롣
|a|=|<a>|

[]

[1]Gθaΰ̿ͭ¤ξ

aΰ̿kȤ롣

a>={e,a,a2,,ak-1}

|<a>|=k

[2]Gθaΰ̵̿¤ξ

a>={,a-2,a-1e,a,a1,a2,}

|<a>|=

󷲤ʬ

[]󷲤ʬϡ󷲤Ǥ롣

[]G󷲡a򤽤Ȥ롣¨G=<a>Ȥ롣

HGʬȤ롣

[1]Hñ̷ξ硢¨H={e}ξ

H={e}=<e>

äơH(eȤ˽󷲤Ȥʤ롣

[2]Hñ̷ǤϤʤ

Hñ̷ǤʤСeʳθʤȤ1H°롣

HθG=<a>θǤ⤢뤿ᡢθaM (M)ηˤʤꡢM0ǤϤʤ(a0=e)

HʬʤΤǡaMεո(aM)-1=a-MH°롣

M0Ǥʤ顢M-MΤɤ餫Ȥ롣

Ĥޤꡢa1,a2,ĤȤGθˡH°ΤɬФƤ롣 ǡa1,a2ǽH°ΤakȤ롣

ޤHʬakHθ顢<ak>θϤ٤Hθˤʤ롣

դˡxHθȤȡHθG=<a>θǤ⤢뤫顢x=anȽ񤱤nϤˡ

ǡnkdzäƾq,;rȤȡn=kq+r, 0rkȤʤ롣

äơ

x
=an
=akq+r
=(ak)qar

x=(ak)qar(*)

ǡ(ak)qHθ顢εոHθǤ롣

(*)κ餫ȡxHθ顢arHθˤʤ롣

rnkdzä;ǡ0rkȤƤ롣

⤷r0ǤϤʤȲꤹȡ0rkȤʤ뤬a1,a2,ĤǺǽH°ΤakȤΤˡarH°ƤޤΤǡ̷⤬롣

äơr=0Ǥ뤿ᡢn=kq+r=kqȤʤ롣

Ĥޤꡢ

x=an=akq(ak)qȤʤꡢ<ak>°뤳Ȥ狼롣

HθϤ٤<ak>°դ<ak>θϤ٤H°ΤǡH=<ak>ȤʤꡢHϽ󷲤Ǥ롣

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