(\mathbb{Q}, +) by |\frac{a}{b}| < 1 a,b\in\mathbb{Z} and b\neq 0 is this a group

**dwsmith** · August 26th 2011, 05:54 PM

$(\mathbb{Q}, +)$ by $\left |\frac{a}{b}\right | < 1, \ a,b\in\mathbb{Z}$ and $b\neq 0$ is this a group.

Identity is 0.
Associative is true.

Inverse (I am not sure but I don't think this works).
If we let $-\frac{a}{b}$ be the inverse, we have a problem since
$\displaystyle\left |\frac{a}{b}\right |=\begin{cases}\frac{a}{b}, \ \frac{a}{b}\geq 0\\-\frac{a}{b}, \ \frac{a}{b}< 0\end{cases}$

So is this not a group or is this approach to the inverse wrong?

**Drexel28** · August 26th 2011, 06:52 PM

Originally Posted by dwsmith

$(\mathbb{Q}, +)$ by $\left |\frac{a}{b}\right | < 1, \ a,b\in\mathbb{Z}$ and $b\neq 0$ is this a group.

Identity is 0.
Associative is true.

Inverse (I am not sure but I don't think this works).
If we let $-\frac{a}{b}$ be the inverse, we have a problem since
$\displaystyle\left |\frac{a}{b}\right |=\begin{cases}\frac{a}{b}, \ \frac{a}{b}\geq 0\\-\frac{a}{b}, \ \frac{a}{b}< 0\end{cases}$

So is this not a group or is this approach to the inverse wrong?

Are you asking whether the set $A=\mathbb{Q}\cap[-1,1]$ is a subgroup of $\mathbb{Q}$ ? Obviously not since $.5+.5=1$ .

**dwsmith** · August 26th 2011, 06:57 PM

Originally Posted by Drexel28

Are you asking whether the set $A=\mathbb{Q}\cap[-1,1]$ is a subgroup of $\mathbb{Q}$ ? Obviously not since $.5+.5=1$ .

I am trying to prove if it is a group or not.

**Drexel28** · August 26th 2011, 06:59 PM

Originally Posted by dwsmith

I am trying to prove if it is a group or not.

And, it's not since it isn't closed under addition.

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