Cia P ua ib pawg Sylow p-sub ntawm G. … Yog G yooj yim, ces nws muaj 10 pawg ntawm kev txiav txim 3 thiab 6 pawg ntawm kev txiav txim 5. Txawm li cas los xij, txij li cov pab pawg no yog txhua lub voj voog ntawm qhov kev txiav txim thawj zaug, txhua yam tsis tseem ceeb ntawm G muaj nyob hauv feem ntau ntawm cov pab pawg no.
Puas yog P pawg cyclic?
Tswv yim pab pawg tsuas yog pawg txiav txim ib leeg xwb, thiab pab pawg C p yog pawg nkaus xwb p.
Puas yog pawg sub cyclic?
Theorem: Tag nrho cov pawg ntawm ib pawg cyclic yog cyclic. Yog tias G=⟨a⟩ yog cyclic, ces rau txhua qhov sib faib d ntawm |G| muaj tseeb ib pawg ntawm kev txiav txim d uas yuav raug tsim los ntawm a|G|/d a | G | /d. Pov thawj: Cia |G|=dn | G |=d n.
Puas P Sylow subgroups normal?
Yog G muaj ib tug Sylow p-subgroup, nws yuav tsum yog ib txwm los ntawm Unique Subgroup of a Given Order is Normal. Piv txwv tias Sylow p-subgroup P yog qhov qub. Tom qab ntawd nws sib npaug nws cov conjugates. Yog li, los ntawm Thib Sylow Theorem, tsuas muaj ib qho xws li Sylow p-subgroup.
Puas yog sylow P-subgroups Abelian?
Peb ua pov thawj tias Sylow p-sub pawg ntawm ib pawg neeg tsis kawg G yog abelian yog tias thiab tsuas yog cov chav kawm loj ntawm p-cov ntsiab lus ntawm G yog tag nrho cov coprime rau p, thiab, yog p ∈ { 3, 5 }, qhov degree ntawm txhua tus cwj pwm irreducible nyob rau hauv tus thawj p-block ntawm G yog coprime rau p.
