$(A\times B)\cap {(C\times D)}=(A\cap C)\times(B\cap D)$ Analogy in Group Theory In set theory we have $(A\times B)\cap {(C\times D)}=(A\cap C)\times(B\cap D)$, is this true in group theory if we replace the $\times$ by products of subgroups i.e.
$$(AB)\cap (CD)=(A\cap C)(B\cap D)$$
I know that $(A\cap C)(B\cap D)$ is contained in $AB\cap CD$ but I could not prove the other inclusion.

No. Take some non trivial group $G$ and consider the product group $G\times G$. Set $A=D=\mathbf{1}\times G$ and $B=C=G\times \mathbf{1}$ where $\mathbf{1}\in G$ is the identity element. Then $AB=G\times G=CD$ and $A\cap C=B\cap D= \mathbf{1}$ so $(A\cap C)(B\cap D)= \mathbf{1}\times\mathbf{1}\neq G\times G=AB\cap CD$.
