# Thread: [SOLVED] Notation: What is this group?

1. ## [SOLVED] Notation: What is this group?

Can someone please tell me what the group $\mathbb{Z}[1/n]$ is?

Never mind - I was having a dull moment. If you are interested, it is just the integers in another guise...

2. Originally Posted by Swlabr
Can someone please tell me what the group $\mathbb{Z}[1/n]$ is?

Never mind - I was having a dull moment. If you are interested, it is just the integers in another guise...

Well, not precisely...if $n=2$ , for example, and since the notation $\mathbb{Z}[1\slash n]$ usually denotes the set of all the values $p(1\slash n)\,,\,\,with\,\,\,p(x)\in\mathbb{Z}[x]$ , then

with $p(x)=x$ we obtain $p(1\slash 2)=1\slash 2\notin\mathbb{Z}$ , so in fact $\mathbb{Z}[1\slash n]\neq \mathbb{Z}$

Tonio

3. Originally Posted by tonio
Well, not precisely...if $n=2$ , for example, and since the notation $\mathbb{Z}[1\slash n]$ usually denotes the set of all the values $p(1\slash n)\,,\,\,with\,\,\,p(x)\in\mathbb{Z}[x]$ , then

with $p(x)=x$ we obtain $p(1\slash 2)=1\slash 2\notin\mathbb{Z}$ , so in fact $\mathbb{Z}[1\slash n]\neq \mathbb{Z}$

Tonio
I took it to be the set $\{\frac{a}{n} \mid a \in \mathbb{Z}\}$ (which is what you are taking it to be, I believe). So when I said "another guise" I meant "isomorphic" not "equal".

$\frac{a}{n} \mapsto a$.