Let $G$ be a group then does $\{e\}/G$ make sense?

Question

For context,

I was trying to show that the quotient of a solvable group is solvable.

For a normal subgroup $N$ I was able to show that ${(G/N)}^{(k)}={G}^{(k)}/N$ and thus by using the fact that $G$ is solvable so ${G}^{(k)}=\{e\}$ for some $k\ge0$ I was trying to show that ${(G/N)}^{(k)}={G}^{(k)}/N=\{N\}$ and then got stuck thinking if $\{e\}/G$ even made any sense.

Answer 1

You don't need to use $G^{(k)}/N$. What is the identity element of that quotient?

You need to show $(G/N)^{(k)}=\{N\}$. What's the homomorphic image of the quotient map, $\phi:G\rightarrow G/N$ for $G^{(k)}$? (Hint: Identity map to identity)