TITLE: What exactly is a coset?
QUESTION [23 upvotes]: If $G$ is a group and $H$ a subgroup of $G$, then the set $gH = \{gh \mid h \in H\} = \{g, gh_1, gh_2,\ldots\}$ is a "left coset" of $G$ wrt $H.$
So does this basically mean, that when I multiply some element of group $H$ on the left by some element from the group $G$, this is a left coset? Doing this multiple times with multiple elements from $G$ and $H$ will give me the set of left cosets? The same for right cosets.
Why is this important then? If the cosets have a one to one relation (like in the Orbit Stabiliser theorem), does that imply a bijection? As one element from the group $G$ is mapped to only another element in $H$ and each of these "outcomes" are unique?

REPLY [8 votes]: Let's go back to basic set theory.  How many elements are in the set:
$$S=\{\{1,2\},\{3,4\}\}$$
If you said $4$, you'd be wrong, there are two - the elements are sets, $\{1,2\}$ and $\{3,4\}$.
While a coset is a set, when we talk about "the set of all cosets," we are actually talking about a set containing sets as members.
You keep talking about individual members of $H$. That is essentially useless. $H$ is a "block" that is acted on uniformly by a single $g$, not individually having different $h\in H$ acted on by different $g\in G$.