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第七章 半群与群 PowerPoint PPT Presentation


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第七章 半群与群. 7.1 半群和独异点的定义及其性质 7.2 半群和独异点的同态与同构 7.3 积半群 7.4 群的基本定义与性质 7.5 置换群和循环群 7.6 子群与陪集 7.7 群的同态与同构. 退出. 7.1 半群和独异点的定义及其性质. 定义 7.1.1 给定 < S ,⊙ > ,若⊙满足结合律,则称 < S ,⊙ > 为半群。 可见,半群就是由集合及其上定义的一个可结合的二元运算组成的代数结构。

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第七章 半群与群

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  • 7.1

  • 7.2

  • 7.3

  • 7.4

  • 7.5

  • 7.6

  • 7.7


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7.1

  • 7.1.1<S><S>

  • 7.1.2<M><M><M>


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  • e<Me>

  • <S>S

  • 7.1.1 <S>(x)(xSxx=x)


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  • 7.1.3<S><S><Me>

  • 7.1.4<S>gSN

  • g<S>:=(x)(xS(n)(nNx=gn))

  • g<S>

  • <Me>gg0=e


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  • 7.1.2

  • <Me>


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  • 7.1.5<S>GS

  • G<S>:=(a)(aSa=(G)) |G|

  • (G)G

  • <Me>


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  • 7.1.6<S>TST<T><S>

  • <Me><Pe>eP


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  • 7.1.3<S>aS<{aa2a3}>

  • a<{aa2a3}><{aa2a3}>


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  • 7.1.4<Me>P<Pe>

  • 7.1.5<Me>


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  • 7.1.6<Me>abMab

  • (1) (a-1)-1=a

  • (2) ab(ab)-1=b-1a-1


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7.2


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  • 7.2.1<S><T>

  • <S><T>:=(f)(fTS(x)( y)(xySf(xy)=f(x)f(y))

  • f<S><T>

  • f


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  • f()f


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  • 7.2.1f<S><T>aSaa=af(a)f(a)=f(a)


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  • 7.2.2g<S><T>h<T><U>hog<S><U>


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  • 7.2.2g<S><S>gg<S><S>g


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  • 7.2.3<S>A={g|g<S><S>}o<Ao>

  • io


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  • 7.2.4<S>B={h|h<S><S>}o<Boi>

  • 7.2.5<S><SSo>SSo<S><SSo>g<S><SSo>


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  • 7.2.3<MeM><TeT>

  • <MeM><TeT>

  • :=(g)(gTM(x)( y)(xyMg(xy)

  • =g(x) g(y))g(eM)=eT

  • g<MeM><TeT>


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  • g(eM)=eT


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  • 7.2.6<M>TMM<M><To>


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7.3


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  • 7.3.1<S><T><ST><S><T>STST

  • <s1t1><s2t2>=<s1s2t1t2>s1s2St1t2T


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  • 7.3.1<S><T><ST>

  • 7.3.2<S><T>e1e2<ST><e1e2><Se1><Te2><ST<e1e2>>


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  • 7.3.3<S><T>12<ST><12>

  • 7.3.4<S><T>ss-1tTt-1<ST><st><s-1t-1>


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7.4

  • 7.4.1<G><G>G<G>


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  • 7.4.2<G>G<G>GG<G>


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  • 7.4.1 <G>|G|1<G>

  • 7.4.2 <G><G>


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  • 7.4.3<G>

  • (a)(b)(c)(abcG((ab=acba=ca)b=c))


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  • 7.4.4<G>

  • (a)(b)(abG((!x)(xGax=b)(!y)(yGya=b))

  • (a)(b)(abG(!x)(!y)(xyG(ax=bya=b))


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  • 7.4.5 <G>(a)(b)(abG(ab)-1=b-1a-1)

  • 7.4.3<G><G><G>Abel

  • 7.4.6<G>

  • <G>Abel(a)(b)(abG(ab)2=a2b2


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  • 7.4.4<G>aGean:=(k)(kI+ {ak=e}=n)aa


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  • 7.4.7<G>aGn

  • (m)(mI+k=mn)ak=e

  • an=end(1dn)ad=ena

  • 7.4.8<G>aGaa-1


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7.5


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  • 7.5.1XXXX|X|

  • X={x1x2xn}n


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  • p-1

  • Xpe


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  • PXX

  • n

  • 7.5.1X={x1x2xn}|PX|=n!


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  • 7.5.2XpipjPXXpipjpipjpipjpipj

  • pipjpiopj

  • pipj=pjopi o

  • xX

  • (pipj)(x)=(pjopi)(x)=pj(pi(x))


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  • 7.5.11-1PX

    • (1)

  • (p1)(p2)(p1,p2PXp1p2PXp2p1PX)

    • (2)

  • (p1)(p2)(p3)(p1,p2,p3PX(p1p2)p3=p1(p2p3))


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    • (3)

  • (pe)(pePX(p)(pPXpep=ppe=p))

    • (4)

  • (p)(pPX(p-1)(p-1PXpp-1=p-1p=pe))

  • (1)PX(2)PX(3)PX(4)PX<PX,><S|X|,>QPX=S|X|Q<Q, >


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    • <S3,>XX<PX,><S|X|,><S3,>X<S3,><S3,>|S3|=3!=6


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    • nn!nSn<Sn, >nn!

    • <{p1,p2},><{p1,p5,p6},>p1,p2,p5,p6S3


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    • 7.5.3<Q>QS|X|R={<ab|abXpQp(a)=b}<Q>X


    4221668

    • 7.5.2<Q>X

    • 7.5.3<G>G


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    • <{e}>

    • ea<{ea}>7.5.27.5.3<{ea}>


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    • <{eab}>7.5.37.5.3

    • AbelAbelAbel


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    • XXXXXTXXXTXXXTXPX

    • (1) (f)(g)(f,gTXfog,gofTX)

    • (2) (f)(g)(h)(f,g,hTX(fog)oh=fo(goh)

    • (3) (i)(iTX(f)(fTXiof=foi=f

    • (4) (f)(fTX(f-1)(f-1TXfof-1=f-1of=i))


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    • <TX,o>

    • TXo<TX,o>


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    • 7.5.4<G>I(g)(gG(a)(aG)(n)(nIa=gn)))<G><G>gg<G>


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    • 7.5.5g<G>I+gn:=(k)(kI+ {gk=e}=n)ng


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    • 7.5.4Abel

    • 7.5.5<G>g|G|=ngn=eG={gg2gn=e} {gk=e}=nn<G>


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    7.6

    • 7.6.1<G>HG<H><H><G><{e}><G><G><G>


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    • 7.6.1 <H><G>eH=eGeHeG<H><G>


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    • 7.6.2<G>HG

    • <H><G>(a)(b)(abH

    • abH)(a)(aHa-1H)

    • <H><G>HH


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    • 7.6.3<G>HG

    • <H><G>(a)(b)(abHab-1H)

    • 7.6.4<G>HG

    • <H><G>(a)(b)(abHabH)


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    • 7.6.2<G>cent Gcent G

    • :={a|aG(x)(xGax=xa)}

    • centGG<G><G>AbelcentG=G

    • 7.6.5 <centG><G>


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    • 7.6.6<G1><G2><G><G1G2><G>

    • 7.6.7<G>


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    • 7.6.3<H><G>aG

    • aH={ah|hH}

    • a<G>HaHaaH


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    • a<G>HHa

    • <G>Abel<H>aH=Ha

    • 7.6.4<G><H>:={<ab|abGb-1aH}


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    • (1) <H><G>H<G>

    • e<G>eH={eh|hH|=H

    • (2) <H><G>aGaaH

    • eHa=aeaH


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    • (3) <H><G>HH

    • f(aH)H

    • f(h)=ah, hH

    • f

    • ah1=ah2h1,h2Hh1=h2f(h1)=f(h2)h1=h2


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    • 7.6.8<H><G>aH=HaH

    • 7.6.9<H><G>aH=bHb-1aH


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    • aHa1

    • a1aHh1Ha1=ah1a-1a1=h1H

    • 7.6.9a1H=aH

    • 7.6.10<H><G>aHbH=aH=bH


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    • GaHaH7.6.10GHHGH

    • 7.6.11<H><G><G>HGGH


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    • <G><H>|G|=n|H|=mGHG=a1Ha2HakHkHGmGkmn=mk(J.L.Lagrange)


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    • 7.6.12<H><G>|G|=n|H|=mn=mkkI+I+


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    • 7.6.5<H><G>GaaH=Ha<H><G>

    • Abel

    • <H>hHah=hah1h2Hah1=h2a


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    • 7.6.13<G><H><G>(a)(aGaHa-1H)


    4221668

    • 7.6.14<G><H>() ( )


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    7.7

    • 7.7.1<G><H><G><H>:=(g)(gHG(a)( b)(abGg(ab)=g(a) g(b)))g<G><H>


    4221668

    • 7.7.1g<G><H>

    • (1) eGeHg(eG)=eH

    • (2) aGg(a-1)=(g(a))-1

    • (3) <S><G>g(S)={g(a)|aS}<g(S)><H>


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    • 7.7.2<G><H>g<G><H><H>

    • g


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    • 7.7.2f<G><H>eH<H>Kf={k|f(k)=eHkG}Kff

    • KfeGKf

    • 7.7.3f<G><H><Kf><G>


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    • 7.7.4f<G><H>fKf={eG}


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    • 7.7.5 f<G><H> fKfa bf(a)=f(b)abG

    • fKffEf


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    • 7.7.6<H><G>g(G/CH)G:g(a)= g<G><G/CH>Kg=


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    • 7.7.7f<G><H> fKf <G><G/ ><G/><H>

    • 10.7.610.7.7


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    • <H><G><G>HG/H:G/H={aH|aG}G/HaHbH=(ab)H

    • <G/H>


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    • 7.7.6*<H><G>g(G/H)G:g(a)=aHg<G><G/H>Kg=H

    • 7.7.7*f<G><H>Kff<G/Kf><H>


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    • <G>f(x)=xxG<G><G><G>?1854A.Cayley

    • 7.7.8


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    • 7.7.9nn


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