Correspondence Theorem (Groups)

Correspondence Theorem (Groups). Let $G$ be a group and let $N \trianglelefteq G$ be a normal subgroup . Let $\pi: G \to G/N$ be the canonical projection. Then the assignments

• $A \mapsto A/N := \{aN : a \in A\}$, for subgroups $A$ with $N \subseteq A \subseteq G$, and
• $B \mapsto \pi^{-1}(B)$ (the preimage ) for subgroups $B \le G/N$,

are inverse, inclusion-preserving bijections between:

1. subgroups $A$ of $G$ with $N \subseteq A$, and
2. subgroups of $G/N$.

Moreover:

• $A \trianglelefteq G$ if and only if $A/N \trianglelefteq G/N$, and
• if $[G:A]$ is finite, then $[G:A] = [G/N : A/N]$.

This theorem explains how the subgroup lattice of a quotient group $G/N$ is “the same as” the lattice of subgroups of $G$ containing $N$. It is a standard tool for building and comparing chains of subgroups, especially in the study of normal series.

Proof sketch. Show $\pi(\pi^{-1}(B)) = B$ and $\pi^{-1}(A/N) = A$ when $N \subseteq A$. Inclusion preservation follows from basic properties of images and preimages, and normality corresponds because conjugation in $G/N$ is induced from conjugation in $G$.