Normal implies modular

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., normal subgroup) must also satisfy the second group property (i.e., modular subgroup)
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Statement

Suppose $A$ is a normal subgroup of $G$. Then, for any subgroup $B$ of $G$ and any subgroup $C$ of $G$ such that $A\le C$, we have:

$A(B\cap C)=AB\cap C$

Facts used

1. Normal implies permutable
2. Permutable implies modular (which in turn uses modular property of groups)

Proof

The proof follows directly by combining facts (1) and (2).