Modularity satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., modular subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

Verbal statement

A modular subgroup of a group is also a modular subgroup inside every intermediate subgroup.

Definitions used

Modular subgroup

Further information: Modular subgroup

A subgroup $A$ of a group $G$ is termed modular in $G$ if for any subgroups $B, C \le G$ such that $A \le C$, we have:

$\langle A, B \cap C \rangle = \langle A,B \rangle \cap C$.

Intermediate subgroup condition

Further information: Intermediate subgroup condition

A subgroup property $p$ is said to satisfy the intermediate subgroup condition if whenever $A \le D \le G$ are groups such that $A$ satisfies property $p$ in $G$, $A$ also satisfies property $p$ in $D$.

Proof

Given: A group $G$, subgroups $A \le D \le G$. $A$ is modular in $G$.

To prove: $A$ is modular in $D$: whenever $B, C \le D$ are such that $A \le C$, we have:

$\langle A, B \cap C \rangle = \langle A, B \rangle \cap C$.

Proof: Since $B,C$ are subgroups of $D$, they are also subgroups of $G$, and the condition $A \le C$ holds by assumption. Thus, by modularity of $A$ in $G$, we conclude that:

$\langle A, B \cap C \rangle = \langle A, B \rangle \cap C$.