If there exist groups $A$ and $B$, as well as their respective normal subgroups $C$ and $D$, then it is possible to prove that $(C\times D) \triangleleft (A\times B)$. I, however, have no idea where to start with this proof. Can someone please give a hint?
2025-01-13 00:08:53.1736726933
Cross-products of groups and their normal subgroups
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Take an arbitrary element of $A \times B$, $(a,b)$ and one of $C \times D$, say $(c,d)$. Compute $(a,b)^{-1}(c,d)(a,b)=(a^{-1},b^{-1})(c,d)(a,b)= \cdots$ and use the fact that $C \unlhd A$ and $D \unlhd B$. Even $(A \times B)$/$(C \times D) \cong$ $A/C$ $\times$ $B/D$.