Contranormality is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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A contranormal subgroup of a contranormal subgroup of a group is contranormal in the whole group.
- Normality satisfies transfer condition: If is normal in , and , then is normal in .
Given: A group , a contranormal subgroup of , a contranormal subgroup of .
To prove: If is a normal subgroup of containing , then .
Proof: By fact (1), is normal in , and contains . Thus, since is contranormal in , . Thus, . So, is a normal subgroup of containing . Since is contranormal in , we get .