Question Number 174879 by mnjuly1970 last updated on 13/Aug/22
$$ \\ $$$$\:\:\:\boldsymbol{\mathrm{Q}}\::\:\left(\boldsymbol{{an}}\:\:\mathscr{E}\:\boldsymbol{{lementary}}\:\boldsymbol{{to}}\:\boldsymbol{{abstract}}\:\boldsymbol{{algebra}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{prove}}\:\boldsymbol{{that}}\:\boldsymbol{{the}}\:\:\boldsymbol{{order}}\:\:\boldsymbol{{of}}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{an}}\:\:\boldsymbol{{element}}\:\:\boldsymbol{{in}}''\:\boldsymbol{{quotient}}\:\boldsymbol{{group}}\:''\:\left(\mathbb{Q}\:,\:\oplus\right)/\left(\mathbb{Z}\:,\:\oplus\right)\:\boldsymbol{{is}}\:\boldsymbol{{finite}}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Notice}}:\:\:\left(\mathbb{Q}\:,\:\oplus\right)/\left(\mathbb{Z}\:,\:\oplus\right)\:=\:\left\{\:\frac{\boldsymbol{{a}}}{\boldsymbol{{b}}}\:+\:\mathbb{Z}\:\mid\:\:\boldsymbol{{a}},\boldsymbol{{b}}\:\in\:\mathbb{Z}\:,\:\boldsymbol{{b}}\:\neq\mathrm{0}\:\right\}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\succ\:\boldsymbol{{Source}}:\:\boldsymbol{{John}}\:\boldsymbol{{B}}\:.\boldsymbol{{F}}\:\boldsymbol{{raleigh}}\:\boldsymbol{{book}}\:\prec\:\:\:\: \\ $$$$ \\ $$
Commented by kaivan.ahmadi last updated on 13/Aug/22
$${let}\:\frac{{a}}{{b}}+\mathbb{Z}\in\frac{{Q}}{\mathbb{Z}}\:\:;{a},{b}\in\mathbb{Z},{b}\neq\mathrm{0}\Rightarrow \\ $$$${b}\ast\left(\frac{{a}}{{b}}+\mathbb{Z}\right)=\left(\frac{{a}}{{b}}+\mathbb{Z}\right)\ast.\underset{{b}−{times}} {.}.\:\ast\left(\frac{{a}}{{b}}+\mathbb{Z}\right) \\ $$$$={a}+\mathbb{Z}\:\underset{\left({a}\in\mathbb{Z}\right)} {=}\mathbb{Z} \\ $$$${since}\:\mathbb{Z}\:{is}\:{the}\:{natural}\:{element} \\ $$$${of}\:\frac{{Q}}{\mathbb{Z}}\:,\frac{{a}}{{b}}+\mathbb{Z}\:{has}\:{finite}\:{order}. \\ $$$$ \\ $$
Commented by mnjuly1970 last updated on 13/Aug/22
$${mercey}\:{master}.. \\ $$
Commented by kaivan.ahmadi last updated on 13/Aug/22
$${thank}\:{you} \\ $$
Commented by kaivan.ahmadi last updated on 13/Aug/22
$${shoma}\:{irani}\:{bodin}? \\ $$
Commented by mnjuly1970 last updated on 14/Aug/22
$$\:{baleh}\:{ostad}\:{arjmand} \\ $$$$\:\:{eradatmandim} \\ $$
Commented by kaivan.ahmadi last updated on 14/Aug/22
$${moafagh}\:{bashi}.\:{lotf}\:{dari}\:{man} \\ $$$${ostad}\:{nistam}. \\ $$
Commented by mnjuly1970 last updated on 14/Aug/22
$${khahesh}\:{mikonam}\:{ghorban} \\ $$$$\:{bozorgvarid}\:...{az}\:{wojood}\: \\ $$$$\:{jenabali}\:{estefadeh}\:{mikonim} \\ $$$$\:\:{zendeh}\:{bashid}\:... \\ $$