I came across with explanation of
For (left)cosets to act like a group, following should be satisfied.
$x \cdot y \in (xN)(yN)$.
i.e $(xN)(yN) = xyN$
And the speaker said “$x$ times $y$ must be in the product of two cosets”
I was confused because as far as I know, the product between two sets $X$ and $Y$ is set of pairs $(x,y)$ with cardinality of card(X) * card(Y) which doesn’t make sense in this context. My question is
What is the definition of “product of $xN$ and $yN$”?
My second question is that the notion “$xyN$“ means normal subgroup N respect to $x \cdot y$ right?
The operation they are referring to is the set-valued product $A\cdot B=\{ab\,:\, a\in A,\, b\in B\}$. The goal is to find conditions on the subgroup $N$ such that the set of left cosets of $N$, endowed with said product, is a group.
It turns out that the condition which guarantees the aforementioned is $N$ being normal. In that case, the group of cosets of $N$, with the projection $\pi:G\to \operatorname{cosets}(G,N)$, $\pi(x)=xN$ is a possible construction for the quotient group $G/N$, i.e. the data of a group $H$ and a map $\pi:G\to H$ such that for all groups $F$ and for all maps $f:G\to F$ such that $\ker f\supseteq N$ there is exactly one map $f':H\to F$ such that $f=f'\circ\pi$.
The notation $xN$, where $x\in G$ and $N<G$, refers the the set $\{xy\,:\, y\in N\}$, which may also be described as $\{g\in G\,:\, x^{-1}g\in N\}$, or as the equivalence class of $x$ with repsect to the equivalence relation $a\sim_N b\iff a^{-1}b\in N$.