TITLE: Is my solution correct? subgroups and quotients of solvable groups are solvable
QUESTION [16 upvotes]: I want to solve the following exercise from Dummit & Foote's Abstract Algebra:

Prove that subgroups and quotient groups of solvable groups are solvable

My attempt:
Let $G$ be a solvable group with a chain $1=G_0 \triangleleft G_1 \triangleleft \dots \triangleleft G_s=G$ such that all factors $G_i/G_{i-1}$ are abelian.

Let $H \leq G$ be a subgroup. We claim that the chain $1=H_0 \leq H_1 \leq \dots \leq H_s=H$ where $H_i=G_i \cap H$ does the trick. Firstly we prove that $H_i \triangleleft H_{i+1}$ for all $i$. Let $x \in H_{i+1}=G_{i+1} \cap H$. Since $x \in H$ we have $x H x^{-1}=H$. Since $x \in G_{i+1} \triangleright G_i$ we have $xG_i x^{-1}=G_i$. Combining these gives $x(H \cap G_i) x^{-1}=H \cap G_i$, as required.
Next, we prove that the quotients $H_{i+1}/H_i=(G_{i+1} \cap H)/(G_i \cap H)$ are abelian. Let $x,y \in G_{i+1} \cap H$. We would like to prove that $xy (G_i \cap H)=yx (G_i \cap H)$. Observe that using the fact the cosets of $G_i$ in $G_{i+1}$ can be interchanged, we find
\begin{equation}
\begin{split}
&xy (G_i \cap H)=(xy G_i) \cap (xy H)=(x G_i) (y G_i) \cap H=(y G_i)(x G_i) \cap (yx H)\\
&=(yx G_i) \cap (yx H)=yx (G_i \cap H).
\end{split}
\end{equation} 

Let $N \trianglelefteq G$ be a normal subgroup, and let $G/N$ be the corresponding quotient group. We suggest the following normal chain:
\begin{equation}
1=G_0/ N  \triangleleft G_1/  N \triangleleft \dots \triangleleft G_s/ N =G/N.
\end{equation}
We first prove that $G_i/N \triangleleft G_{i+1}/N$:
Let $gN \in G_{i+1}/N,aN \in G_i/N$. We have $(gN)(aN)(gN)^{-1}=(gag^{-1})N$. Since $G_i \trianglelefteq G_{i+1}$ the element $gag^{-1} \in G_i$, which gives $(gN)(G_i/N)(gN)^{-1} \in G_i/N$ for all $gN \in G_{i+1}/N$.
The quotients (or factors) are
\begin{equation}
(G_{i+1}/N) \big/ (G_i/N),
\end{equation}
and we wish to prove they are abelian. Indeed, let $g_1N(G_iN),g_2 N(G_iN) \in (G_{i+1}/N) \big/ (G_i N)$ (i.e. $g_1,g_2 \in G_{i+1}$). These two arbitrary elements commute iff
\begin{equation}
[g_2Ng_1N]^{-1}[g_1Ng_2N] \in G_i/N
\end{equation}
which is equivalent to the condition $g_1^{-1}g_2^{-1}g_1g_2 \in G_i$. Since we have that the quotient $G_{i+1}/G_i$ is abelian we know that $(g_1G_i)(g_2G_i)=(g_2G_i)(g_1 G_i)$ which gives precisely this condition. We conclude that quotient groups of solvable groups are solvable. 

Is my solution correct? If not, please help me fix it. Thanks!

REPLY [10 votes]: As Eric Auld mentioned in the comment, the statement that $G_0/N=1/N$ is incorrect. To fix this, you may use the normal chain
$$\begin{equation}
1=G_0N/ N  \triangleleft G_1N/  N \triangleleft \dots \triangleleft G_sN/ N =G/N.
\end{equation}$$