Third isomorphism theorem, about quotients

Question

I'm trying to understand the third isomorphism theorem statement, specifically the one in Wikipedia: https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem I'm stuck at the point (1) because I don't agree with the statement, it doesn't make sense to me that:

If $K$ is a subgroup of $G$ such that $N \subseteq K \subseteq G$, then $K/N$ is a subgroup of $G/N$.

The single fact that bugs me is that you can't say that $K/N$ is a subgroup of $G/N$, because to be a subgroup you must be subset, and satisfy certain axioms... but $K/H$ it's not even a subset of G/H? I don't see how one is a subset of the other. We know $K$ is a subset of $G$, that doesn't imply, to me, that $K/N$ is a subset of $G/N$, in any obvious way. I did a lot of exercises about groups, I need help understanding this fact.

Sorry for the latex, I really don't know it, I just copied the phrase on wikipedia, I promiss I will learn when I have some time.

Answer 1

Your issue seems to be with definitions and notation. It is always important to remember what the symbols you are working with stand for!

In general, given a group $H$ and a subgroup $N$, we define $$H/N = \{ h + N \, | \,h \in H\},$$ where $$h + N$$ is the left coset of $N$ corresponding to $h$, defined as $$h + N= \{h +n \, | \, n \in N \}.$$ The set $H/N$ is the set of all cosets (cosets are defined for any subgroup $N$, although the interested cases occur when $N$ is normal).

Now apply this definition to the groups in question: we want to look at the cosets of $N$ in both $G$ and $K \subset G$.

We get that

$$G/N = \{ g+ N \, | \, g \in G\},$$ and

$$K/N = \{ k + N \, | \, k \in K \} \subset G/N,$$ where the containment holds because all cosets $ k +N$ for $k \in K$ also define cosets in $G$, since $k \in K \subset G$.

Answer 2

$G/N$ is the set of all cosets of the form $Ng$ where $g \in G$. $K/N$ is the set of all cosets of the form $Nk$ where $k \in K$.

Since $K \subseteq G$, then $k \in K$ implies $k \in G$.
It follows that $Nk \in G/N$.
Hence $K/N \subseteq G/N$.

Now, if $K/N$ and $G/N$ are both groups, then we must have $K/N$ is a subgroup of $G/N$.

The third isomorphism theorem tells you that this is true when $N \subseteq K \subseteq G$ and $N$ is a normal subgroup of $G$.