Txoj Kev Qhia Ib Pawg yog Abelian
- Show the commutator [x, y]=xyx−1y−1 [x, y]=x y x − 1 y − 1 of two arbitary elements x, y∈G x, y ∈ G yuav tsum yog tus kheej.
- Qhia cov pab pawg isomorphic rau cov khoom ncaj qha ntawm ob pawg abelian (sub).
Yuav ua li cas koj thiaj paub yog ib pab pawg sib tham?
Yog hais tias txoj cai sib pauv tau tuav hauv ib pawg, ces ib pab pawg no hu ua Abelian pawg lossis pawg sib txuas. Yog li cov pab pawg (G, ∗) tau hais tias yog pawg Abelian lossis pawg sib txuas yog tias a∗b=b∗a, ∀a, b∈G. Ib pawg uas tsis yog Abelian hu ua pawg uas tsis yog Abelian.
Koj ua li cas qhia ib pab pawg tsis yog Abelian?
Definition 0.3: Pawg Abelian Yog ib pab pawg muaj cov cuab yeej uas ab=ba rau txhua khub ntawm cov ntsiab lus a thiab b, peb hais tias pawg yog Abelian. Ib pawg tsis yog-Abelian yog tias muaj qee yam ntawm cov ntsiab lus a thiab b uas ab=ba.
Dab tsi ua rau pawg tsis yog-Abelian?
Nyob rau hauv lej, thiab tshwj xeeb hauv pab pawg kev xav, ib pab pawg tsis yog abelian, qee zaum hu ua pawg tsis sib txuas, yog ib pawg (G, ∗) uas muaj tsawg kawg ib khub ntawm cov ntsiab lus a thiab b ntawm G, xws li a ∗ b ≠ b ∗ a Cov pab pawg no sib txawv nrog pawg abelian.
Puas yog txhua pab pawg neeg abelian?
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. Hauv ib pab pawg Abelian, txhua lub ntsiab lus yog nyob rau hauv chav kawm conjugacy los ntawm nws tus kheej, thiab lub cim lub rooj koom nrog lub hwj chim ntawm ib lub ntsiab lus hu ua pab pawg neeg tsim khoom.
