# Contranormality is transitive

From Groupprops

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)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about contranormal subgroup |Get facts that use property satisfaction of contranormal subgroup | Get facts that use property satisfaction of contranormal subgroup|Get more facts about transitive subgroup property

## Statement

### Verbal statement

A contranormal subgroup of a contranormal subgroup of a group is contranormal in the whole group.

## Facts used

- Normality satisfies transfer condition: If is normal in , and , then is normal in .

## Proof

**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 .