Namely, I have the notation, for groups $G,H$, that $\frac{G}{H}$.
The details is that ,
Let $G,H,A$ be groups. Then If $H \triangleleft G$ and $A \subseteq G$ then $H \cap A \triangleleft A$ and $\frac{A}{H \cap A} \cong \frac{HA}{H}$
Now, apparently, this is a restatement of the first isomoprhism theorem for groups,
Which I am familiar with in the form
Let $G,H$ be groups and a homomorphism $\theta:G \to H$ with kernel $N=ker(\theta)$ then $N \triangleleft G$ and with $Im(\theta) \leq H$, there is an isomoprhism $\phi$
$$\phi: G/N \to Im(\theta)$$
So, questions
How is the first statement saying the same thing as the second? Is $H:=N$? But even so, why are there three groups $G,H,A$? The second statement only has two.
Now, $G/H$ is the quotient group(i.e. the left/right cosets of $H$). But $\frac{G}{H}$? What's that? $G,H$ are...well, "sets" with an operation yes? Purely seeing them as a set of elements doesn't make sense to literally "divide."
Very confused. The first statement in from chapter $14$ of Ian Stewart's Galois theory 4th edition by the way. Second from personal lecture notes in uni.
Can someone clear up the two questions for me please?
And here's another
If $H \triangleleft G$ and $H \subseteq A \triangleleft G$ then $H \triangleleft A$, $A/H \triangleleft G/H$ and
$$\frac{G/H}{A/H} \cong \frac{G}{A}$$
is apparently the second isomorphism theorem. Does the last one mean...$(G/H)/(A/H)$?
