Group Isomorphisms. Need clarification of question.

Question

Let $A,B,C$ be subgroups of a group $D$ such that $B,C \trianglelefteq D$ and $A=B\cap C$. Show that $$ \frac{D/B}{C/A} \cong \frac{D/C}{B/A}$$ My question is simply a clarification of what $\frac{D/B}{C/A}$ and $\frac{D/C}{B/A}$ are. Specifically, isn't the quotient group $\frac{D/B}{C/A}$ defined only when $C/A \trianglelefteq D/B$, but $C/A$ is not even a subset of $D/B$.

Answer 1

Let $f$ be the composition of the following two maps: $$C \xrightarrow{\;\;\text{inclusion}\;\;} D\xrightarrow{\;\;\text{quotient}\;\;} D/B$$ Note that $\ker(f)=B\cap C=A$, so by the first isomorphism theorem there is an injective induced map $$C/A\xrightarrow{\;\;\text{induced}\;\;} D/B$$ The notation $\dfrac{D/B}{C/A}$ implicitly identifies $C/A$ with its image under this injective map.