Normality is not transitive and prime index only forces it for the smallest prime

1. A.If $G_{1}$ is normal in G, then $(G_{2}G_{1})/G_{1} \cong G_{2}/(G_{2} \cap G_{1})$.✓ Correct
2. B.If $H_{1}$ and $H_{2}$ are normal subgroups of $G_{1}$ and $G_{2}$, respectively, then $(G_{1} \times G_{2})/(H_{1} \times H_{2}) \cong (G_{1}/H_{1}) \times (G_{2}/H_{2})$.✓ Correct
3. C.If $G_{1}$ is normal in $G_{2}$ and $G_{2}$ is normal in G, then $G_{1}$ is normal in G.
4. D.Every subgroup of prime index in G is normal.

The trap it tests

A property assumed to pass to subobjects, quotients, or through a chain. It does not.