Subgroup correspondence for a surjective group homomorphism ($B⟷{\theta }^{-1}(B),A⟷\theta (A)(K\le A)$)

For a surjection $\theta :G\to H$ with kernel $K$, inverse image and image give inverse order-preserving bijections between subgroups of $H$ and subgroups of $G$ containing $K$. They preserve index. For an arbitrary subgroup $A$, ${\theta }^{-1}(\theta (A))=AK$, so $[H:\theta (A)]=[G:AK]$.

// Target: algebra.bigb