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Q-an-E-lementary-to-abstract-algebra-prove-that-the-order-of-an-element-in-quotient-group-Q-Z-is-finite-Notice-Q-Z-a-b-




Question Number 174879 by mnjuly1970 last updated on 13/Aug/22
     Q : (an  E lementary to abstract algebra )            prove that the  order  of              an  element  in” quotient group ” (Q , ⊕)/(Z , ⊕) is finite.            Notice:  (Q , ⊕)/(Z , ⊕) = { (a/b) + Z ∣  a,b ∈ Z , b ≠0 }                                 ≻ Source: John B .F raleigh book ≺
Q:(anElementarytoabstractalgebra)provethattheorderofanelementinquotientgroup(Q,)/(Z,)isfinite.Notice:(Q,)/(Z,)={ab+Za,bZ,b0}Source:JohnB.Fraleighbook
Commented by kaivan.ahmadi last updated on 13/Aug/22
let (a/b)+Z∈(Q/Z)  ;a,b∈Z,b≠0⇒  b∗((a/b)+Z)=((a/b)+Z)∗.._(b−times) . ∗((a/b)+Z)  =a+Z =_((a∈Z)) Z  since Z is the natural element  of (Q/Z) ,(a/b)+Z has finite order.
letab+ZQZ;a,bZ,b0b(ab+Z)=(ab+Z)..btimes.(ab+Z)=a+Z=(aZ)ZsinceZisthenaturalelementofQZ,ab+Zhasfiniteorder.
Commented by mnjuly1970 last updated on 13/Aug/22
mercey master..
merceymaster..
Commented by kaivan.ahmadi last updated on 13/Aug/22
thank you
thankyou
Commented by kaivan.ahmadi last updated on 13/Aug/22
shoma irani bodin?
shomairanibodin?
Commented by mnjuly1970 last updated on 14/Aug/22
 baleh ostad arjmand    eradatmandim
balehostadarjmanderadatmandim
Commented by kaivan.ahmadi last updated on 14/Aug/22
moafagh bashi. lotf dari man  ostad nistam.
moafaghbashi.lotfdarimanostadnistam.
Commented by mnjuly1970 last updated on 14/Aug/22
khahesh mikonam ghorban   bozorgvarid ...az wojood    jenabali estefadeh mikonim    zendeh bashid ...
khaheshmikonamghorbanbozorgvaridazwojoodjenabaliestefadehmikonimzendehbashid

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