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

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?

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 $$Na$$.