Contranormality is transitive

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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

  1. Normality satisfies transfer condition: If L is normal in G, and K \le G, then K \cap L is normal in K.

Proof

Given: A group G, a contranormal subgroup K of G, a contranormal subgroup H of K.

To prove: If L is a normal subgroup of G containing H, then L = G.

Proof: By fact (1), L \cap K is normal in K, and contains H. Thus, since H is contranormal in K, L \cap K = K. Thus, K \le L. So, L is a normal subgroup of G containing K. Since K is contranormal in G, we get L = G.