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

\boldsymbol Q : (\boldsymbol a n E \boldsymbol l e m e n t a r y \boldsymbol t o \boldsymbol a b s t r a c t \boldsymbol a l g e b r a)

\boldsymbol p r o v e \boldsymbol t h a t \boldsymbol t h e \boldsymbol o r d e r \boldsymbol o f

\boldsymbol a n \boldsymbol e l e m e n t \boldsymbol {i n}^{″} \boldsymbol q u o t i e n t \boldsymbol g r o u p^{″} (Q, \oplus) / (Z, \oplus) \boldsymbol i s \boldsymbol f i n i t e .

\boldsymbol N o t i c e : (Q, \oplus) / (Z, \oplus) = {\frac{\boldsymbol a}{\boldsymbol b} + Z ∣ \boldsymbol a, \boldsymbol b \in Z, \boldsymbol b \neq 0}

≻ \boldsymbol S o u r c e : \boldsymbol J o h n \boldsymbol B . \boldsymbol F \boldsymbol r a l e i g h \boldsymbol 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 \dots 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 \dots