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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A modular subgroup of a group is also a modular subgroup inside every intermediate subgroup.
Further information: Modular subgroup
A subgroup of a group is termed modular in if for any subgroups such that , we have:
Intermediate subgroup condition
Further information: Intermediate subgroup condition
A subgroup property is said to satisfy the intermediate subgroup condition if whenever are groups such that satisfies property in , also satisfies property in .
Related subgroup properties satisfying intermediate subgroup condition
- Permutability satisfies intermediate subgroup condition
- Normality satisfies intermediate subgroup condition
- Ellipticity satisfies intermediate subgroup condition
Given: A group , subgroups . is modular in .
To prove: is modular in : whenever are such that , we have:
Proof: Since are subgroups of , they are also subgroups of , and the condition holds by assumption. Thus, by modularity of in , we conclude that: