A general idea of $G/H$

Question

I am looking t see whether my understanding is correct. SO for a group $G$ and its subgroup $H$, $G/H$ is another group called the quotient group.

Now, there are a few things that revolve around this object, which is my source of confusion and difficulty in digesting it fully like equivalence relation, equivalence class, cosets and normal groups. Say an example in my lecture note says

"we define $G/H := \{\text{all left/right cosets of $H$ over $G$}\}$ where $H$ is normal"

If $H$ is normal then the right cosets are precisely the left cosets. So okay, that's some way to define the quotient group but my question is

Is that the general way to define $G/H$? That they are the set of left cosets or right cosets?

Well if $H$ is not normal, then ambiguity arises in whether we want left or right cosets. So I am guessing not.

Take $\mathbb{Z}$. Then $\mathbb{Z}_{12}$ is, notation wise, equivalent to $\mathbb{Z}/12\mathbb{Z}$. Now, I know that $\mathbb{Z}_{12}=\{0,1,...,11\}$. But $\mathbb{Z}/12\mathbb{Z}$, is this equivalent to that under addition?

Using the notion in normal groups, I need to consider left or right cosets of $12\mathbb{Z}=\{0, \pm 12, \pm 24, \pm 36,...\}$ which gives,(just taking left for this case)

$0+12\mathbb{Z}=\{0, \pm 12, \pm 24, \pm 36,...\}$

$1+12\mathbb{Z}=\{1, 13, -11, 25,...\}$

$2+12\mathbb{Z}=\{2, 14, -10, 26, ...\}$

and so on. Have I made basic errors so far?

Now, I want to check if I am right; when I get to $12+12 \mathbb{Z}$, I will get $12 \mathbb{Z}$. So are these $n+12\mathbb{Z}$ precisely the equivalence classes of $n$ under modulo $12$?

If so, then did this work because $n+12\mathbb{Z}=12\mathbb{Z}+n$ i.e. normal in $\mathbb{Z}$? How about for groups that are not normal? Can I define them as the first boxed statement? Or, will it have to be some other equivalence relation aptly defined? Namely,

Is $G/H$ defined as

1. $G/H := \{\text{all left/right cosets of $H$ over $G$}\}$ where $H$ is a subgroup(not necessarily normal in $G$)

2. or just as "set of equivalence classes for equivalence relations defined accordingly and not necessarily that of (1.)"?

Answer 1

For any subgroup $H$ of any group $G$, $G/H$ is the set of left cosets, while $H\backslash G$ is the set of right cosets. They are different if $H$ is not a normal subgroup. But the map $gH \mapsto Hg^{-1}$ makes a bijection between them, taking the left $G$-action to the right one.

Answer 2

For subgroups $H$ which are not normal, the set of left cosets $G/H$ cannot be given a group structure deduced from the group structure of $G$. For this left cosets must be the same as right cosets (incidentally, the set of right cosets modulo $H$ is denoted $H\backslash G$).