find the the order of the element $\frac{2}{3}+\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$.

Question

i have some confusion.,,,that this question has already asked here how to find the order of an element in a quotient group

Let $\mathbb{Q}/\mathbb{Z}$ be the quotient group of the additive group of rational numbers.then the order of the element $\frac{2}{3}+\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$.

$1) 2 $

$2) 3 $

$c) 4$

$d) 6$

My attempts : $\frac{\mathbb{Q}}{\mathbb{Z}}$ additive groups of rational numbers

$(2/3 + \mathbb{Z} ) + (2/3 + \mathbb{Z}) + (2/3 + \mathbb{Z})= 2 + \mathbb{Z}=\mathbb{Z}$

$(2/3 + \mathbb{Z} ) + (2/3 + \mathbb{Z}) + (2/3 + \mathbb{Z}) +(2/3 + \mathbb{Z} ) + (2/3 + \mathbb{Z}) + (2/3 + \mathbb{Z})=6 + \mathbb{Z}=\mathbb{Z}$

now my answer is option $ 2)$ and option $4)$

But according to duplicate Question answer is given $3$ that order will be $3$

im confusing that why order $6$ is not correct ???

Any hints/solutiuon will be appreciated

Answer 1

The order of an element $x$ in a group $G$ (let's say abelian, written additively) is defined to be the smallest positive integer $n$ for which $n\cdot x=0$. In your case, we have $G=\Bbb Q/\Bbb Z$ and $x=\frac23+\Bbb Z$, and you've noted that $3\cdot x=0$ and $6\cdot x=0$. Since $3$ is smaller than $6$, and no smaller positive integer satisfies $n\cdot x=0$, we have that the order of $\frac23+\Bbb Z$ is $3$.

Answer 2

Remember that in general, the order of an element $g$ in a group $G$ is the smallest integer $n$ such that $g^n = e$ or ($ng = 0$ in additive notation. )

Answer 3

An element in a group only has one order (the question also talks about "the order"): the smallest integer $n$ such that $ng=0$ (writing the group operation additively).

With your interpretation and your example, e.g. $9$ (any multiple of $3$) would be an order but not among the options, making the options poorly chosen.

Answer 4

You have to choose the smallest one because order is the least positive integer $n$ such that $a^n =e$.

Now here $3$ is the least positive integer .it is not only $6$ but also for any multiple of $3$ you get.$a^{3n}$$=$$e$.(Where $e$ is the identity element).

In $\mathbb{Q} / \mathbb{Z}$ for any element $p/q + \mathbb{Z}$ the order of that element is $q$.Where $gcd(p,q)=1$.