Txhua pab pawg ntawm pawg neeg abelian yog ib txwm, yog li txhua pab pawg ua rau nce mus rau ib pawg neeg. Subgroups, quotients, thiab ncaj qha sums ntawm abelian pawg yog dua abelian. Cov pab pawg abelian yooj yim kawg nkaus yog cov pab pawg cyclic ntawm kev txiav txim thawj zaug.
Vim li cas txhua pab pawg ntawm pawg Abelian ib txwm muaj?
(1) Txhua pawg ntawm pawg Abelian yog ib txwm vim ah=ha rau txhua tus ∈ G thiab rau tag nrho h ∈ H. (2) Qhov chaw Z(G) ntawm ib pawg ib txwm ib txwm muaj txij li ah=ha rau txhua tus ∈ G thiab rau tag nrho h ∈ Z(G).
Puas yog txhua pab pawg ntawm pawg Abelian cyclic?
Txhua pawg neeg caij tsheb kauj vab yog Abelian, tab sis ib pab pawg Abelian tsis tas yuav mus ncig. … Txhua pawg ntawm pawg Abelian yog ib txwm muaj. Nyob rau hauv ib pab pawg Abelian, txhua lub ntsiab yog nyob rau hauv ib chav kawm conjugacy los ntawm nws tus kheej, thiab lub cim lub rooj koom nrog lub hwj chim ntawm ib lub caij hu ua ib pab pawg neeg generator.
Puas yog ib pawg pab pawg Abelian ib txwm?
Ua pov thawj tias ib pab pawg ntawm pawg Abelian yog ib pawg sub. Teb: Rov qab: Ib pawg H ntawm ib pawg G hu ua ib txwm yog gH=Hg rau txhua g ∈ G. … gh=hg rau tag nrho h vim G yog Abelian. Yog li ntawd {gh | h ∈ H }={hg | h ∈ H}=Hg by definition of right coset Hg.
Puas yog txhua pab pawg ib txwm muaj?
Txhua pab pawg yog ib pab pawg ntawm nws tus kheej. Ib yam li ntawd, pawg tsis tseem ceeb yog ib pawg ntawm txhua pawg.). Ntawm no, qhov thib ob yog qhov qub tab sis thawj tsis yog.
