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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-potentially relatively characteristic subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications |Get help on looking up subgroup property implications/non-implications
Get more facts about normal subgroup| Get more facts about normal-potentially relatively characteristic subgroup

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Contents 1 Statement 2 Facts used 3 Proof 3.1 Example of the dihedral group

Statement

A normal subgroup of a group need not be a normal-potentially relatively characteristic subgroup.

Facts used

1. Normal not implies normal-extensible automorphism-invariant
2. Normal-potentially relatively characteristic

Proof

The proof follows directly from facts (1) and (2).

Example of the dihedral group

Further information: dihedral group:D8

Let G be the dihedral group of order eight, and H be one of the Klein four-subgroups.

• H is not a normal-potentially characteristic subgroup of G: Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of G can be extended to an automorphism of K for any group K containing G as a normal subgroup. But since there is an automorphism of G not sending H to itself, H is not invariant under this automorphism, which extends to an automorphism of K.
• H is normal in G: This is obvious.