Permutable implies modular
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., permutable subgroup) must also satisfy the second subgroup property (i.e., modular subgroup)
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Any permutable subgroup of a group is modular.
Further information: Permutable subgroup
A subgroup of a group is termed permutable if for every subgroup .
Further information: Modular subgroup
- Modular property of groups: This states that if are subgroups of such that , then:
Given: A subgroup of a group such that for all subgroups .
To prove: For any subgroups of such that , we have:
Proof: Since is permutable, we have:
, so .
Applying fact (1) now yields the result.