Clarification of definition of quotient group - Does $Na, a \in G$ mean "cosets of $N$ in $G$"?

Question

The definition of quotient group from mathworld.wolfram.com is:

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$, written $G/N$ and read "G modulo N", is the set of cosets of $N$ in $G$.

I don't understand the part "set of cosets of $N$ in $G$". Later it continues:

The elements of $G/N$ are written $Na$.

From this, I conclude that the expression "set of cosets of $N$ in $G$" means the set of cosets which have elements $Na, a \in G$. Is this correct?

Answer 1

The notation "$Na$" means exactly "the coset of $a$ in $G/N$."

So $G/N$ is exactly the collection of cosets $\{Na\mid a\in G\}$. Note that this set notation does not irredundantly list the cosets.

I don't think "the set of cosets which have elements $Na, a \in G$" is a valid way of saying it. The cosets don't have $Na$ as elements, each coset is an $Na$.