Modularity satisfies intermediate subgroup condition

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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.

Related facts

Related subgroup properties satisfying intermediate subgroup condition

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.