From what I understand, to find the quotient group requires knowing all cosets of a subgroup. I know what a coset is defined, but when it comes to infinite groups, I'm a bit confused. For example,
Here's a group $[R;+]$, and one of its normal subgroup $[Z;+]$. Find all cosets of $Z$ in $R$.
I was thinking:
1. For $0 \in R$, $0+Z=Z+0=Z$, so $Z$ itself is a coset.
2. For $0.1 \in R$, we get a group $G=\{\cdots,0.1,1.1,2.1,\cdots\}$.
Clearly $G \ne Z$, so $G$ is a different coset from $Z$. Things will get similar if I do it by $0.01 \in R$. It seems that I actually cannot list all cosets of $Z$ or sepecify what these cosets are like. But then how should I describe the quotient group (of $R$) if I cannot write down what these cosets are?
Here's another example which I can specify cosets:
Here's a group $[R^*:=\{x\ne 0 \mid x \in R\};\cdot]$ and one of its normal subgroup $[R^+:=\{x>0\mid x \in R\}]$. Find all cosets of $R^+$ in $R^*$.
$R^*$ can be partitioned into positive real numbers $R^+$ and negative real numbers $R^-$.
1. If a real number $r>0$, $rR^+=R^+r=R^+$;
2. If a real number $r<0$, $rR^+=R^+r=R^-$.
Now we say $[\{R^+,R^-\};\cdot]$ is a quotient group of $R^+$ in $R^*$.
Go back to the first example and even further: under a more general circumstance, what should I do to find the quotient group?
Any kind help would be appreciated.
Let $G=(\Bbb R, +)$ and $H=(\Bbb Z, +)$. Then the (left) cosets of $H$ in $G$ are of the form $$a+H:=\{a+h\mid h\in H\},$$ where $a$ is in the interval $[0,1)$.
(The right cosets are defined dually.)