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 ≺


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Question Number 174879 by mnjuly1970 last updated on 13/Aug/22

$Q : (a n E l e m e n t a r y t o a b s t r a c t a l g e b r a)$ $p r o v e t h a t t h e o r d e r o f$ $a n e l e m e n t {i n}^{″} q u o t i e n t g r o u p^{″} (Q, \oplus) / (Z, \oplus) i s f i n i t e .$ $N o t i c e : (Q, \oplus) / (Z, \oplus) = {\frac{a}{b} + Z ∣ a, b \in Z, b \neq 0}$ $≻ S o u r c e : J o h n B . F r a l e i g h b o o k ≺$
Commented by kaivan.ahmadi last updated on 13/Aug/22

$l e t \frac{a}{b} + Z \in \frac{Q}{Z}; a, b \in Z, b \neq 0 \Rightarrow$ $b * (\frac{a}{b} + Z) = (\frac{a}{b} + Z) * . \underset{b - t i m e s}{.} . * (\frac{a}{b} + Z)$ $= a + Z \underset{(a \in Z)}{=} Z$ $s i n c e Z i s t h e n a t u r a l e l e m e n t$ $o f \frac{Q}{Z}, \frac{a}{b} + Z h a s f i n i t e o r d e r .$
Commented by mnjuly1970 last updated on 13/Aug/22

$m e r c e y m a s t e r . .$
Commented by kaivan.ahmadi last updated on 13/Aug/22

$t h a n k y o u$
Commented by kaivan.ahmadi last updated on 13/Aug/22

$s h o m a i r a n i b o d i n ?$
Commented by mnjuly1970 last updated on 14/Aug/22

$b a l e h o s t a d a r j m a n d$ $e r a d a t m a n d i m$
Commented by kaivan.ahmadi last updated on 14/Aug/22

$m o a f a g h b a s h i . l o t f d a r i m a n$ $o s t a d n i s t a m .$
Commented by mnjuly1970 last updated on 14/Aug/22

$k h a h e s h m i k o n a m g h o r b a n$ $b o z o r g v a r i d . . . a z w o j o o d$ $j e n a b a l i e s t e f a d e h m i k o n i m$ $z e n d e h b a s h i d . . .$
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