# Find the quotient group of an infinite subgroup.

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.

• Since there are uncountably many such cosets in the first example, it seems a little unreasonable to ask you to find all of them! They probably intend you to give a uniform description. For example each coset has a unique representative $x$ satisfying $0 \le x < 1$. Commented Mar 16, 2019 at 5:16
• Are $R$ the reals and $Z$ the integers? Commented Mar 16, 2019 at 6:16
• @B.Swan yes they are. Commented Mar 16, 2019 at 7:48
• @DerekHolt Thanks. But I wonder: Is it the only way to solve this problem that I try every $g \in G$ to see whether $Ag(A \leqslant G)$ is a distinct coset? Commented Mar 16, 2019 at 7:54
• In the first case, as stated above, your cosets are $xZ$ with $0\leq x <1$, can you see why all of them are different and why those are the only ones? In general you need to pick an arbitrary $x$ from a group and see how $xZ$ looks like. Then you need to see which elements induce different cosets. Commented Mar 16, 2019 at 7:55

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 $$[0,1)$$.